21 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
1 A line through the origin ★★★
Let \(W=\{(x,y)\in\mathbb{R}^2 : y=3x\}\). Is \(W\) a subspace of \(\mathbb{R}^2\)? Justify.
2 A half-plane ★★★
Let \(H=\{(x,y)\in\mathbb{R}^2 : x\ge 0\}\). Show that \(H\) is not a subspace.
3 Two pairs of vectors ★★★
Decide whether each pair is linearly independent: (a) \((1,2)\) and \((2,4)\); (b) \((1,2)\) and \((3,1)\).
4 Counting dimensions ★★★
Give the dimension of each space: (a) \(\mathbb{R}^4\); (b) \(P_2\), the polynomials of degree at most 2; (c) \(M_{2\times3}\); (d) the diagonal \(3\times3\) matrices.
5 Is it linear? ★★★
(a) Show that \(T(x,y)=(2x+y,\,x-y)\) is linear and find its matrix. Compute \(T(1,2)\). (b) Show that \(S(x,y)=(x+1,\,y)\) is not linear.
6 Rank-nullity quick check ★★★
A \(4\times6\) matrix has rank 3. Find the dimension of its null space and say in which \(\mathbb{R}^k\) its column space lives.
7 Coordinates in a basis ★★★
Let \(B=\{(1,0),(1,1)\}\). Find the coordinates of \(v=(4,3)\) in \(B\).
8 Basis of a solution set ★★★
Let \(W=\{(x,y,z) : x+y+z=0 \text{ and } x-z=0\}\). Find a basis of \(W\) and its dimension.
9 An affine plane ★★★
Explain why \(\{(x,y,z): x+y+z=1\}\) is not a subspace of \(\mathbb{R}^3\), in two different ways.
10 Polynomial basis ★★★
Show that \(p_1=1+x\), \(p_2=x+x^2\), \(p_3=1+x^2\) form a basis of \(P_2\).
11 In the span or not? ★★★
Let \(u=(1,1,1)\) and \(w=(0,1,2)\). Decide whether (a) \((3,5,7)\) and (b) \((3,5,8)\) lie in \(\operatorname{span}\{u,w\}\).
12 Null and column space ★★★
Let \(B=\begin{pmatrix}1&0&2\\0&1&-1\\1&1&1\end{pmatrix}\). Find a basis of \(\mathrm{Nul}(B)\) and of \(\mathrm{Col}(B)\), and check rank-nullity.
13 A rotation ★★★
The map \(T\) rotates the plane by \(90^\circ\) counterclockwise. Find its matrix, compute \(T(3,5)\), then \(T(T(3,5))\).
14 Kernel and range ★★★
Let \(T(x,y,z)=(x-y,\,y-z)\). Find a basis of \(\ker T\) and decide whether \(T\) is onto \(\mathbb{R}^2\).
15 Trace-zero matrices ★★★
Let \(W\) be the set of \(2\times2\) matrices with trace 0. Show that \(W\) is a subspace of \(M_{2\times2}\), find a basis, and give \(\dim W\).
16 The derivative as a linear map ★★★
Let \(D:P_3\to P_2\) be \(D(p)=p'\). Show that \(D\) is linear, find \(\ker D\), and use rank-nullity to show \(D\) is onto.
17 Coordinates in a new basis ★★★
Let \(B=\{(1,1),(1,-1)\}\). Write \(P\), compute \(P^{-1}\), and find \([v]_B\) for \(v=(7,3)\).
18 Matrix in a basis ★★★
Let \(T(x,y)=(x+2y,\,3x+2y)\) and \(B=\{(1,1),(1,-1)\}\). Find \([T]_B\) and verify that trace and determinant agree with those of the standard matrix.
19 Preserving independence ★★★
Suppose \(v_1,v_2,v_3\) are linearly independent. Prove that \(v_1+v_2\), \(v_2+v_3\), \(v_1+v_3\) are also linearly independent.
20 A parameter and dependence ★★★
For which value of \(a\) are \((1,1,0)\), \((1,2,1)\), \((2,a,1)\) linearly dependent? Give the dependence relation.
21 Recovering a map from a basis ★★★
A linear map \(T:\mathbb{R}^2\to\mathbb{R}^2\) satisfies \(T(1,1)=(3,1)\) and \(T(1,-1)=(1,5)\). Find its standard matrix and compute \(T(4,-2)\).
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