These notes accompany MATH 146, the honours stream of linear algebra at the University of Waterloo. The course covers the same ground as MATH 136 but moves faster, proves more, and insists on the abstract viewpoint from the outset. We work over the field \(\mathbb{R}\) of real numbers throughout, though almost everything generalises to any field.

Chapter 1: Vectors and Geometry in \(\mathbb{R}^n\)

The concrete world of \(\mathbb{R}^n\) is where linear algebra begins, and where geometric intuition is richest. Even after we pass to abstract vector spaces in Chapter 3, we will return here repeatedly to ground our ideas. The goal of this chapter is to establish \(\mathbb{R}^n\) as a vector space, equip it with the dot product, and see how orthogonality and projection organise geometry.

1.1 \(\mathbb{R}^n\) as a Vector Space

Before giving the abstract definition of a vector space, it is instructive to observe that \(\mathbb{R}^n\) with its familiar operations already satisfies every algebraic law we could want. This serves as both motivation and warm-up.

These two operations interact in exactly the right ways. One verifies that \(\mathbb{R}^n\) satisfies the ten axioms V1–V10 that will define a vector space in Chapter 3 (commutativity, associativity, zero vector, additive inverse, distributivity, and the scalar-one axiom). A sum \(c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k\) is called a linear combination of \(\mathbf{v}_1, \ldots, \mathbf{v}_k\).

This simple observation is the engine behind our ability to strip spanning sets down to their essentials.

Linear Independence

A spanning set may contain redundancy — some vectors expressible in terms of the others. Linear independence is the condition that no such redundancy exists.

Notice that the set is dependent if and only if at least one vector can be written as a linear combination of the others — the two formulations are equivalent, and it is worth making sure you can pass fluently between them.

Basis and Standard Basis

Affine Subsets: Lines, Planes, and \(k\)-Planes

Geometry in \(\mathbb{R}^n\) is organised by these flat, translated subspaces.

1.2 Dot Product and Norms

The dot product promotes \(\mathbb{R}^n\) from a mere vector space to a space with angles and distances. This additional structure is what makes geometry possible.

The dot product is symmetric (\(\mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x}\)), bilinear (\(\mathbf{x} \cdot (s\mathbf{y} + t\mathbf{z}) = s(\mathbf{x} \cdot \mathbf{y}) + t(\mathbf{x} \cdot \mathbf{z})\)), and positive definite (\(\mathbf{x} \cdot \mathbf{x} \geq 0\) with equality iff \(\mathbf{x} = \mathbf{0}\)). These three properties are what make it an inner product.

Setting \(t = (\mathbf{x} \cdot \mathbf{y})/\|\mathbf{y}\|^2\) and simplifying yields the result. \(\square\)

The Cauchy–Schwarz inequality justifies the definition of the angle \(\theta \in [0,\pi]\) between nonzero vectors by \(\cos\theta = (\mathbf{x} \cdot \mathbf{y})/(\|\mathbf{x}\|\|\mathbf{y}\|)\), since the right-hand side lies in \([-1,1]\).

1.3 Orthogonality and Projections

Two vectors are orthogonal if \(\mathbf{x} \cdot \mathbf{y} = 0\), i.e., if the angle between them is \(\pi/2\). Projection is the operation that decomposes a vector into its component along a given direction and its component perpendicular to it.

One checks directly that \(\operatorname{proj}_{\mathbf{v}}\mathbf{u}\) is a scalar multiple of \(\mathbf{v}\) and that \(\operatorname{perp}_{\mathbf{v}}\mathbf{u}\) is orthogonal to \(\mathbf{v}\), so the decomposition \(\mathbf{u} = \operatorname{proj}_{\mathbf{v}}\mathbf{u} + \operatorname{perp}_{\mathbf{v}}\mathbf{u}\) is an orthogonal direct sum. Geometrically, \(\operatorname{proj}_{\mathbf{v}}\mathbf{u}\) is the “shadow” of \(\mathbf{u}\) cast on the line through \(\mathbf{v}\).

Chapter 2: Systems of Linear Equations

Solving a system of linear equations is the most concrete problem in the subject, and the algorithmic answer — Gaussian elimination to row echelon form — is the workhorse of computational linear algebra. But the real payoff of this chapter is theoretical: the rank of a matrix, the uniqueness of RREF, and a first glimpse of the dimension theorem.

2.1 Row Operations and Echelon Forms

The rank of a matrix \(A\), written \(\operatorname{rank}(A)\), is the number of leading ones in its RREF.

RREF Uniqueness

The uniqueness of RREF is not obvious — different sequences of row operations might, in principle, produce different reduced forms. The following theorem says they cannot.

Proof. Suppose \(R\) and \(R'\) are both RREFs of \(A\). Since they are row equivalent to each other, they represent the same linear map \(\mathbb{R}^n \to \mathbb{R}^m\) (up to a change of basis in the domain given by elementary matrices, which are invertible). More concretely, let \(j_1 < j_2 < \cdots < j_r\) be the pivot columns of \(R\) and \(j_1' < \cdots < j_{r'}'\) the pivot columns of \(R'\). Since row operations preserve the solution space of the homogeneous system, we have \(\operatorname{Null}(R) = \operatorname{Null}(R')\), hence \(r = r'\). A careful induction on the pivot columns, using the fact that the free variables of \(R\) and \(R'\) parametrise the same nullspace, shows \(j_k = j_k'\) for all \(k\) and moreover that the non-pivot entries in each pivot row are forced to be equal. Hence \(R = R'\). \(\square\)

Rank and the Solution Count

2.2 The Rank-Nullity Theorem (Preview)

The \(n - r\) free variables in a consistent system index a basis for the solution set when the system is homogeneous. Since the solution space of \(A\mathbf{x} = \mathbf{0}\) is the nullspace \(\operatorname{Null}(A)\), and the \(r\) pivot columns span the columnspace \(\operatorname{Col}(A)\), we get a preview of the main structural theorem:

The full theorem and its abstract form will appear in Chapter 4.

2.3 Geometric Interpretation

Geometrically, each linear equation \(a_1 x_1 + \cdots + a_n x_n = b\) cuts out a hyperplane in \(\mathbb{R}^n\). The solution set of a system is the intersection of \(m\) hyperplanes. A consistent system has a solution set that is either a point (\(r = n\)) or an affine \((n-r)\)-plane. An inconsistent system corresponds to hyperplanes with empty intersection — a phenomenon that cannot occur in \(\mathbb{R}^1\) but is generic in higher dimensions.

Chapter 3: Abstract Vector Spaces

One of the great insights of modern mathematics is that the algebraic structure behind \(\mathbb{R}^n\) — its axioms — appears in wildly different settings: spaces of functions, spaces of matrices, spaces of polynomials. By working with the axioms directly, we prove theorems once and apply them everywhere.

3.1 Axioms and Examples

Notice that V4 and V5 make \((V, +)\) an abelian group, and the remaining axioms say scalar multiplication is a ring homomorphism \(\mathbb{R} \to \operatorname{End}(V)\) compatible with the group structure.

Proof. For (3): \(0\mathbf{v} = (0+0)\mathbf{v} = 0\mathbf{v} + 0\mathbf{v}\); adding \(-0\mathbf{v}\) to both sides gives \(\mathbf{0} = 0\mathbf{v}\). \(\square\)

3.2 Subspaces

Rather than verifying all ten axioms for a subset, we have a simpler test.

Proof. The “only if” direction is immediate. For “if”: closure under scalar multiplication with \(c = 0\) gives \(\mathbf{0} = 0\mathbf{u} \in W\), so V4 holds; taking \(c = -1\) gives additive inverses, so V5 holds. The remaining axioms hold in \(W\) because they hold in the larger space \(V\). \(\square\)

The key examples: the span of any set of vectors is a subspace; the solution set of a homogeneous linear system is a subspace; the set of polynomials with a prescribed root is a subspace of \(\mathcal{P}\).

3.3 Linear Independence and Span

The definitions of span and linear independence carry over from \(\mathbb{R}^n\) word-for-word.

A key structural lemma clarifies the relationship between span and independence.

Proof. By induction we construct spanning sets \(S_0 = \{\mathbf{w}_1, \ldots, \mathbf{w}_n\}\), \(S_1, \ldots, S_m\) where \(S_k\) contains \(\mathbf{u}_1, \ldots, \mathbf{u}_k\) and \(n - k\) of the original \(\mathbf{w}_j\)’s, and still spans \(V\). At step \(k\), write \(\mathbf{u}_k\) as a linear combination of elements of \(S_{k-1}\); since \(\mathbf{u}_k \notin \operatorname{Span}\{\mathbf{u}_1, \ldots, \mathbf{u}_{k-1}\}\) (by independence), at least one \(\mathbf{w}_j\) appears with a nonzero coefficient, and we replace that \(\mathbf{w}_j\) by \(\mathbf{u}_k\). After \(m\) steps we have not run out of \(\mathbf{w}_j\)’s, so \(m \leq n\). \(\square\)

The Steinitz Exchange Lemma is the spine of dimension theory: it immediately implies that all bases of \(V\) have the same size.

3.4 Bases and Dimension

Proof. If \(\mathcal{B}\) and \(\mathcal{B}'\) are both bases, then \(\mathcal{B}\) is linearly independent and \(\mathcal{B}'\) spans, so \(|\mathcal{B}| \leq |\mathcal{B}'|\) by Steinitz. Symmetrically \(|\mathcal{B}'| \leq |\mathcal{B}|\). \(\square\)

Quotient Spaces

The quotient construction is an essential tool in functional analysis and in the first isomorphism theorem.

These operations are well-defined (independent of coset representative) precisely because \(W\) is closed under addition and scalar multiplication.

Proof. If \(\{\mathbf{v}_1 + W, \ldots, \mathbf{v}_k + W\}\) is a basis for \(V/W\) and \(\{\mathbf{w}_1, \ldots, \mathbf{w}_m\}\) is a basis for \(W\), then \(\{\mathbf{v}_1, \ldots, \mathbf{v}_k, \mathbf{w}_1, \ldots, \mathbf{w}_m\}\) is a basis for \(V\), giving \(\dim V = k + m\). \(\square\)

3.5 Coordinates

Once a basis is fixed, every vector has a unique coordinate representation, and the abstract vector space becomes isomorphic to \(\mathbb{R}^n\).

The map \(\mathbf{v} \mapsto [\mathbf{v}]_{\mathcal{B}}\) is linear and bijective — a vector space isomorphism \(V \xrightarrow{\sim} \mathbb{R}^n\). Given two bases \(\mathcal{B}\) and \(\mathcal{C}\), the change-of-basis matrix \({}_{\mathcal{C}}P_{\mathcal{B}}\) has columns \([\mathbf{v}_i]_{\mathcal{C}}\) and satisfies \([\mathbf{x}]_{\mathcal{C}} = {}_{\mathcal{C}}P_{\mathcal{B}}[\mathbf{x}]_{\mathcal{B}}\). Note \(({}_{\mathcal{C}}P_{\mathcal{B}})^{-1} = {}_{\mathcal{B}}P_{\mathcal{C}}\).

Chapter 4: Linear Transformations

A vector space without maps between vector spaces is like a set without functions — a static object. Linear transformations are the structure-preserving maps of the category of vector spaces, and they are the central objects of linear algebra.

4.1 Definition and Examples

Notice that a linear map is completely determined by its values on any basis: if \(\mathcal{B} = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\) is a basis for \(V\) and \(\mathbf{w}_1, \ldots, \mathbf{w}_n \in W\) are arbitrary, there exists a unique linear \(T : V \to W\) with \(T(\mathbf{v}_i) = \mathbf{w}_i\). This is the universal property of a basis, and it is extraordinarily useful.

4.2 The Kernel and Image

The two fundamental subspaces attached to a linear map encode everything about its injectivity and surjectivity.

Proof. If \(T\) is injective and \(T(\mathbf{v}) = \mathbf{0} = T(\mathbf{0})\), then \(\mathbf{v} = \mathbf{0}\). Conversely, if \(\ker T = \{\mathbf{0}\}\) and \(T(\mathbf{u}) = T(\mathbf{v})\), then \(T(\mathbf{u} - \mathbf{v}) = \mathbf{0}\), so \(\mathbf{u} - \mathbf{v} \in \ker T = \{\mathbf{0}\}\), giving \(\mathbf{u} = \mathbf{v}\). \(\square\)

The Four Fundamental Subspaces

For a matrix \(A \in M_{m \times n}(\mathbb{R})\), the associated linear map \(\mathbf{x} \mapsto A\mathbf{x}\) gives four subspaces: the column space \(\operatorname{Col}(A) = \operatorname{im}(A) \subseteq \mathbb{R}^m\), the null space \(\operatorname{Null}(A) = \ker(A) \subseteq \mathbb{R}^n\), the row space \(\operatorname{Row}(A) = \operatorname{Col}(A^T) \subseteq \mathbb{R}^n\), and the left null space \(\operatorname{Null}(A^T) \subseteq \mathbb{R}^m\). Strikingly:

Proof. Write \(\mathbf{a} = A^T\mathbf{y}\). Then \(\mathbf{a} \cdot \mathbf{x} = (A^T\mathbf{y})^T\mathbf{x} = \mathbf{y}^T(A\mathbf{x}) = \mathbf{y}^T\mathbf{0} = 0\). \(\square\)

4.3 Rank-Nullity Theorem

This is the central theorem of the chapter — indeed, one of the most important theorems in all of linear algebra.

Proof. Let \(\{\mathbf{k}_1, \ldots, \mathbf{k}_s\}\) be a basis for \(\ker T\); extend to a basis \(\{\mathbf{k}_1, \ldots, \mathbf{k}_s, \mathbf{v}_1, \ldots, \mathbf{v}_r\}\) for \(V\), so \(\dim V = r + s\). We claim \(\{T(\mathbf{v}_1), \ldots, T(\mathbf{v}_r)\}\) is a basis for \(\operatorname{im} T\). Spanning: any \(T(\mathbf{u})\) with \(\mathbf{u} = \sum a_i\mathbf{k}_i + \sum b_j\mathbf{v}_j\) satisfies \(T(\mathbf{u}) = \sum b_j T(\mathbf{v}_j)\). Independence: if \(\sum b_j T(\mathbf{v}_j) = \mathbf{0}\), then \(T(\sum b_j\mathbf{v}_j) = \mathbf{0}\), so \(\sum b_j\mathbf{v}_j \in \ker T = \operatorname{Span}\{\mathbf{k}_i\}\), forcing all \(b_j = 0\). Hence \(\operatorname{rank}(T) = r\) and \(\operatorname{nullity}(T) = s\). \(\square\)

The rank-nullity theorem says that a linear map “uses up” dimension: the more it collapses (large kernel), the less it maps onto (small image). The two always add up to the dimension of the domain.

Isomorphism Theorems

Proof. Define \(\tilde{T} : V/\ker T \to \operatorname{im} T\) by \(\tilde{T}(\mathbf{v} + \ker T) = T(\mathbf{v})\). This is well-defined (cosets of \(\ker T\) have the same \(T\)-image), linear, surjective onto \(\operatorname{im} T\), and injective (if \(T(\mathbf{v}) = \mathbf{0}\) then \(\mathbf{v} \in \ker T\), so the coset is zero in \(V/\ker T\)). \(\square\)

4.4 Matrix Representations

Every linear map between finite-dimensional spaces can be encoded as a matrix, once bases are chosen.

The key property: \([T(\mathbf{v})]_{\mathcal{C}} = [T]_{\mathcal{B}}^{\mathcal{C}} [\mathbf{v}]_{\mathcal{B}}\). For a linear operator \(T : V \to V\), choosing \(\mathcal{B} = \mathcal{C}\) gives the \(\mathcal{B}\)-matrix \([T]_{\mathcal{B}}\). If \(P = {}_{\mathcal{E}}\!P_{\mathcal{B}}\) is the change-of-basis matrix from \(\mathcal{B}\) to the standard basis \(\mathcal{E}\), then \([T]_{\mathcal{B}} = P^{-1}[T]_{\mathcal{E}}P\) — a key formula for diagonalisation.

Chapter 5: Dual Spaces

Every vector space \(V\) carries a shadow: its dual space, the space of all linear functionals on \(V\). The dual is not exotic — it is just the space of linear maps \(V \to \mathbb{R}\) — but it leads to a surprisingly rich and symmetric theory, and it is the right language for understanding transposes, bilinear forms, and later, tensors.

5.1 Linear Functionals and the Dual Space

Since \(\mathbb{R}\) is itself a vector space, \(V^*\) is a vector space over \(\mathbb{R}\). Examples: (a) for any \(\mathbf{a} \in \mathbb{R}^n\), the map \(\phi_{\mathbf{a}}(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x}\) is a linear functional on \(\mathbb{R}^n\); (b) evaluation \(\text{ev}_t : \mathcal{P} \to \mathbb{R}\), \(p \mapsto p(t)\) is a linear functional; (c) integration \(\int_0^1 : C([0,1]) \to \mathbb{R}\) is a linear functional.

5.2 Dual Basis

Proof. Spanning: for any \(\psi \in V^*\), let \(c_i = \psi(\mathbf{v}_i)\). Then \(\psi - \sum c_i \phi^i\) vanishes on each \(\mathbf{v}_j\), hence on all of \(V\), so \(\psi = \sum c_i\phi^i\). Independence: if \(\sum c_i\phi^i = 0\), evaluate on \(\mathbf{v}_j\) to get \(c_j = 0\). \(\square\)

The dual basis \(\phi^i\) is often written as a row vector: \(\phi^i\) extracts the \(i\)-th coordinate with respect to \(\mathcal{B}\). Notice that \(\phi^i(\mathbf{v}) = [\mathbf{v}]_{\mathcal{B},i}\) is the \(i\)-th component of the coordinate vector.

5.3 The Transpose of a Linear Map

The notation is natural: \(T^*\) “pulls back” functionals from \(W\) to \(V\). The matrix of \(T^*\) with respect to the dual bases is the transpose of the matrix of \(T\):

Proof. Compute: \((T^*\phi^j)(\mathbf{v}_i) = \phi^j(T(\mathbf{v}_i)) = \phi^j\!\left(\sum_k a_{ki}\mathbf{w}_k\right) = a_{ji}\). Thus \(T^*\phi^j = \sum_i a_{ji}\psi^i\), meaning the \(j\)-th column of \([T^*]\) is the \(j\)-th row of \(A\), i.e., \([T^*] = A^T\). \(\square\)

This explains why the matrix transpose is the “right” operation on matrices: it corresponds to taking the dual of the associated linear map.

Chapter 6: Determinants

Determinants are simultaneously a computational tool (testing invertibility, computing volumes) and a conceptual cornerstone (characterising alternating multilinear forms). The cleaner approach — and the one that makes all the properties obvious — is to start from the algebraic definition.

6.1 Multilinear Alternating Forms

Think of a square matrix as \(n\) column vectors. A function of those \(n\) vectors that is linear in each separately, and changes sign when two columns are swapped, is called an alternating multilinear form.

Proof (sketch). If \(D\) is alternating multilinear and \(\mathbf{A} = [\mathbf{a}_1 \cdots \mathbf{a}_n]\), write each \(\mathbf{a}_j = \sum_i a_{ij}\mathbf{e}_i\). By multilinearity, \(D(\mathbf{A}) = \sum_\sigma a_{\sigma(1),1}\cdots a_{\sigma(n),n} D(\mathbf{e}_{\sigma(1)}, \ldots, \mathbf{e}_{\sigma(n)})\) where the sum runs over all permutations \(\sigma\). The alternating property forces \(D(\mathbf{e}_{\sigma(1)}, \ldots, \mathbf{e}_{\sigma(n)}) = \operatorname{sgn}(\sigma) D(\mathbf{e}_1, \ldots, \mathbf{e}_n)\), so \(D = D(I) \cdot \det\) where \(\det\) is the unique alternating multilinear form normalised by \(\det(I) = 1\). \(\square\)

6.2 Properties and Computation

The defining properties immediately yield the row-operation rules that make computation tractable.

Consequently: \(\det A = 0\) if \(A\) has two equal rows, or a zero row, or a row that is a linear combination of the others.

Proof. Both sides define alternating multilinear functions of the columns of \(AB\). Since the space of such forms is one-dimensional, they are proportional; evaluating at \(A = I\) gives proportionality constant \(\det(B)\), yielding \(\det(AB) = \det(A)\det(B)\). \(\square\)

Proof. \(A\) is invertible iff its RREF is \(I\), iff its rows are linearly independent. Row-reduce \(A\) to \(I\); each ERO multiplies the determinant by a nonzero scalar, so \(\det(A) \neq 0\) iff \(\det\) was never forced to zero, iff no row became zero during elimination, iff the rows are independent. \(\square\)

The adjugate matrix is \(\operatorname{adj}(A) = (\operatorname{cof} A)^T\), and \(A^{-1} = \frac{1}{\det A}\operatorname{adj}(A)\) for invertible \(A\).

6.3 Geometric Interpretation: Volume

The determinant measures signed volume. This is the deepest geometric content of the subject.

In dimension 2, \(|\det[\mathbf{u}\ \mathbf{v}]|\) is the area of the parallelogram spanned by \(\mathbf{u}\) and \(\mathbf{v}\). The sign records orientation: positive if the ordered pair \((\mathbf{u}, \mathbf{v})\) is a positively oriented frame, negative otherwise. The formula \(\det(AB) = \det(A)\det(B)\) then says: a linear transformation multiplies all volumes by the factor \(|\det A|\).

Chapter 7: Eigenvalues and Diagonalization

Given a linear operator \(T : V \to V\), can we find a basis in which the matrix of \(T\) is diagonal — i.e., a basis of “eigen-directions”? This question leads to the spectral theory of matrices and is fundamental to differential equations, quantum mechanics, Markov chains, and data science.

7.1 Characteristic Polynomial

For a matrix \(A\), this means \((A - \lambda I)\mathbf{v} = \mathbf{0}\) has a nonzero solution, which happens iff \(\det(A - \lambda I) = 0\).

Notice that \(\operatorname{tr}(A) = \lambda_1 + \cdots + \lambda_n\) (sum of eigenvalues) and \(\det(A) = \lambda_1 \cdots \lambda_n\) (product of eigenvalues), where the \(\lambda_i\) are the roots of \(p_A\) counted with multiplicity. Similar matrices have the same characteristic polynomial, so eigenvalues are properties of the operator, not the matrix.

7.2 Eigenspaces and Independence

Proof. By induction on \(k\). The base case \(k=1\) is clear (\(\mathbf{v}_1 \neq \mathbf{0}\)). Suppose the result holds for \(k-1\), and suppose \(\sum_{i=1}^k c_i\mathbf{v}_i = \mathbf{0}\). Apply \(A\) to get \(\sum_{i=1}^k c_i\lambda_i\mathbf{v}_i = \mathbf{0}\), then subtract \(\lambda_k\) times the original: \(\sum_{i=1}^{k-1}c_i(\lambda_i - \lambda_k)\mathbf{v}_i = \mathbf{0}\). By induction, \(c_i(\lambda_i - \lambda_k) = 0\) for \(1 \leq i \leq k-1\); since \(\lambda_i \neq \lambda_k\), we get \(c_i = 0\), and then \(c_k = 0\) follows. \(\square\)

7.3 Diagonalizability Criterion

Proof. \(A\) is diagonalizable iff \(\mathbb{R}^n\) has a basis of eigenvectors of \(A\). By Theorem 7.6, bases for distinct eigenspaces are independent; collecting them gives a basis for \(\mathbb{R}^n\) iff their total size \(\sum g_{\lambda_i} = n = \sum a_{\lambda_i}\), which requires both splitting and \(g_\lambda = a_\lambda\) for all \(\lambda\). \(\square\)

7.4 Powers of Matrices and the Cayley–Hamilton Theorem (Preview)

If \(A = PDP^{-1}\), then \(A^k = PD^kP^{-1}\) and \(D^k = \operatorname{diag}(\lambda_1^k, \ldots, \lambda_n^k)\). This makes computing large powers of diagonalisable matrices essentially free.

The Cayley–Hamilton theorem will be proved in full in MATH 245 using the Smith normal form or the theory of modules over a PID. Here we note a corollary: every power \(A^k\) for \(k \geq n\) can be expressed as a linear combination of \(I, A, A^2, \ldots, A^{n-1}\), so the algebra \(\mathbb{R}[A]\) is at most \(n\)-dimensional. This is the beginning of the theory of the minimal polynomial, which will tell us exactly when a matrix is diagonalizable and what Jordan normal form looks like.

Appendix: The Invertible Matrix Theorem

The following compendium shows how many apparently different conditions on a square matrix are in fact equivalent. Let \(A\) be an \(n \times n\) real matrix.

Each of these conditions is a different lens on the same phenomenon: a square matrix either establishes a perfect bijection between \(\mathbb{R}^n\) and itself, or it collapses some nonzero subspace to zero. There is no middle ground.