Practice: Linear algebra

Linear algebra

Warm-upHuman

What is the rank of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & 1 \end{pmatrix}$ ?

Linear algebra

Warm-upHuman

Which of these are subspaces of the given vector space? (Choose all that apply.)

Choose all that apply

\{(t, 2t) : t \in \R\} \subseteq \R^2

\{(x, y, z) : x + y + z = 1\} \subseteq \R^3

\{x \in \R^n : Ax = 0\}

for a fixed

m \times n

matrix

A

\{(x, y) : xy = 0\} \subseteq \R^2

The polynomials of degree exactly

2

, inside all polynomials

Linear algebra

CoreHuman

A linear map $T : \R^7 \to \R^4$ is surjective. What is $\dim \ker T$ ?

Linear algebra

CoreHuman

$A$ is a $3 \times 3$ matrix with $\det A = 5$ . What is $\det(2A)$ ?

Linear algebra

CoreHuman

Which of these real matrices are diagonalisable over $\R$ ? (Choose all that apply.)

Choose all that apply

\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}

\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}

\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}

Linear algebra

CoreHuman

In the proof that eigenvectors $v_1, \dots, v_m$ with distinct eigenvalues are independent, we reach $c_i(\lambda_i - \lambda_m) = 0$ for $i < m$ . Which fact lets us conclude $c_i = 0$ ?

Choose one

\lambda_i \neq \lambda_m

, and in a field a product of two non-zero elements is non-zero

v_i \neq 0

The induction hypothesisLinearity of

T

Linear algebra

StretchHuman

For a linear map $T : V \to V$ we proved: if $V$ is finite-dimensional, then $T$ injective $\Rightarrow$ $T$ surjective. Which example shows that the finite-dimensionality hypothesis cannot be dropped?

Choose one

On the space of all real sequences,

T(a_0, a_1, a_2, \dots) = (0, a_0, a_1, \dots)

On all polynomials,

T = d/dx

On

\R^2

,

T(x, y) = (x, 0)

On

\R^2

, rotation by

90^\circ

Linear algebra

CoreFormal

Let f : V →ₗ[K] W with [FiniteDimensional K V]. Which Lean statement correctly formalises the rank–nullity theorem?

Choose one

finrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K Vfinrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K Wfinrank K (LinearMap.range f) ≤ finrank K Vfinrank K (LinearMap.ker f) = finrank K V - finrank K W

Linear algebra

StretchComputational

Write rank(rows) that returns the rank of a matrix (a list of rows of integers) by Gaussian elimination with exact Fraction arithmetic. Then use it to check rank–nullity on the photosynthesis matrix: its null space should be one-dimensional.

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