Exercises

Exercise 3.1

(See Solution 3.1.) Let \(V = \{ (x, y, z) \ | \ x, y, z \in {\bf R} \}\). (Thus, \(V = {\bf R}^3\).) We use the regular addition of vectors. However, in contrast to the regular scalar multiplication (Definition 3.7), we now use the following. Decide in each case whether this turns \(V\) into a vector space:

\(r \cdot (x, y, z) = (rx, y, rz)\),
\(r \cdot (x, y, z) = (0,0,0)\),
\(r \cdot (x,y,z) = (r^2x, r^2y, r^2z)\).

Exercise 3.2

(See Solution 3.2.) Let \(V \subset {\bf R}^2\) be a subspace. Which of the following statements are correct?

\(V\) contains at least one element.
\(V\) contains at least two elements.
\(V\) contains the zero vector \((0,0)\).
If \(v, w \in V\) then also \(v - w \in V\).

Exercise 3.3

(See Solution 3.3.) Using basic properties of differentiable functions from your calculus class, show that the space

\[ \{ f : {\bf R} \to {\bf R} \ | \ f \text{ is differentiable}\} \]

is a vector space (with the sum and scalar multiple defined as in Equation (3.23) and Equation (3.24)).

Hint: structure your thinking as in Definition and Lemma 3.22.

Exercise 3.4

(See Solution 3.4.) Give an example of two subspaces \(V, W \subset {\bf R}^2\) such that their union

\[ V \cup W = \{ x = (x_1, x_2) \in {\bf R}^2 \ | \ x \in V \text{ or } x \in W \} \]

is not a subspace.

Hint: Example 3.13.

Also give an example of two subspaces \(V, W \subset {\bf R}^2\), where the union \(V \cup W\) is a subspace.

Hint: be very lazy and minimalistic. What is the smallest subspace you can come up with?

Exercise 3.5

(See Solution 3.5.) Determine in each case whether \(w \in {\bf R}^4\) lies in the span of \(v_1\) and \(v_2\). If so, name at least one linear combination of \(v_1\) and \(v_2\) that equals \(w\); otherwise explain why there is no such linear combination.

\(w = (2, -1,0,1)\), \(v_1 = (1,0,0,1)\), \(v_2 = (0,1,0,1)\)
\(w = (1,2,15,11)\), \(v_1 = (2,-1,0,2)\), \(v_2=(1,-1,-3,1)\)
\(w = (2,5,8,3)\), \(v_1 = (2,-1,0,5)\), \(v_2 = (-1, 2, 2, -3)\)

Exercise 3.6

(See Solution 3.6.) Determine whether the following vectors span \({\bf R}^4\):

\((1,1,1,1)\), \((0,1,1,1)\), \((0,0,1,1)\), \((0,0,0,1)\)
\((1,3,-5,0)\), \((-2, 1,0,0)\), \((0,2,1,-1)\), \((1, -4, 5, 0)\)

Exercise 3.7

(See Solution 3.7.) Determine whether the following vectors are linearly independent:

\(v_1 = (1, -1, 0)\), \(v_2=(3,2,-1)\), \(v_3 = (3,5,-2)\) in \(V = {\bf R}^3\),
\(v_1 = (1,1,1)\), \(v_2 = (1,-1,1)\), \(v_3 = (0,0,1)\) in \(V = {\bf R}^3\),
\((1,-1,1,-1)\), \((2,0,1,0)\), \((0,-2,1,-2)\) in \({\bf R}^4\),
\((1,1,0,0)\), \((1,0,1,0)\), \((0,0,1,1)\) and \((0,1,0,1)\) in \({\bf R}^4\).

Exercise 3.8

(See Solution 3.8.) Name three vectors \(v_1, v_2, v_3 \in {\bf R}^2\) such that:

\(v_1, v_2\) are linearly independent,
\(v_1, v_3\) are linearly independent, and
\(v_2, v_3\) are linearly independent, but
\(v_1, v_2, v_3\) are not linearly independent.

Exercise 3.9

(See Solution 3.9.) Consider the vector space \(V = {\bf R}[x]^{\le 3}\) of polynomials of degree at most 3. Decide which of the following subsets of \(V\) is a subspace:

\(\{ f \ | \ f \in V, f(2) = 1 \}\),
\(\{ x \cdot f \ | \ f \in {\bf R}[x]^{\le 2} \}\),
\(\{ x \cdot f + (1-x) g \ | \ f, g \in {\bf R}[x]^{\le 2} \}\),
\(\{ f \ | \ f\in {\bf R}[x]^{\le 3}, f(0) = 0\}\).

Exercise 3.10

(See Solution 3.10.) Express the following polynomials as linear combinations of \(x+1\), \(x-1\) and \(x^2-1\) (in \({\bf R}[x]^{\le 2})\): \(x^2 + 4x-2\), \(x\), \(40-x^2\).

Exercise 3.11

(See Solution 3.11.) Is the following sentence correct? “In \({\bf R}[x]^{\le 3}\), the polynomial \(f(x) = \frac 1 4 x^3 + 3x +1\) is a linear combination of the polynomials \(x^2\), \(x\) and \(1\) since \(f(x) = \frac x 4 \cdot x^2 + 3 \cdot x + 1\).”

Exercise 3.12

(See Solution 3.12.) Express each of the three standard basis vectors \(e_1, e_2, e_3\) as a linear combination of the basis vectors in Example 3.63.

Exercise 3.13

(See Solution 3.13.) Consider \(A = \left ( \begin{array}{cc} 1 & 1 \\ 2 & 2 \end{array} \right )\) and \(B = \left ( \begin{array}{cc} 3 & 2 \\ 3 & 5 \end{array} \right )\) in the vector space of \(2 \times 2\)-matrices. Is \(C = \left ( \begin{array}{cc} -1 & 0 \\ 2 & 4 \end{array} \right )\) a linear combination of \(A\) and \(B\)?

Exercise 3.14

(See Solution 3.14.) In the vector space \({\mathrm {Mat}}_{2 \times 3}\) of \(2 \times 3\)-matrices, we consider the set

\[ T = \left \{ \left ( \begin{array}{ccc} x_1 & x_2 & x_3 \\ x_4 & x_5 & x_6 \end{array} \right ) \ | \ x_1 + x_4 + x_6 = 0, x_1 + x_4 + x_3 + x_5 = 0 \right \}. \]

Decide whether \(T\) is a subspace of \({\mathrm {Mat}}_{2 \times 3}\).
Find all the vectors (i.e., matrices) in \(T\).
Find some vectors such that \(T = L(v_1, v_2, v_3, v_4)\).

Exercise 3.15

(See Solution 3.15.) In \({\bf R}^4\) consider the subset

\[ S = \{(x,y,z,t) \ | \ x+y+z+t = 0 \}. \]

Decide whether \(S\) is a subspace of \({\bf R}^4\).
Find all the vectors in \(S\).
Find some vectors such that \(S = L(v_1, v_2, v_3)\).

Exercise 3.16

(See Solution 3.16.) Consider the following two subspaces of \({\bf R}^4\):

\[ S = L((1,-1,0,1), (2,1,-2,0),(0,0,1,1)) \]

and \(T\), which is the solution set of the system

\[ \begin{align*} 2x_1-x_2-3x_4 & = 0 \\ 2x_1 + x_3 + x_4 & = 0. \end{align*} \]

Determine \(S \cap T\).

Exercise 3.17

(See Solution 3.17.) Consider the following two subspaces of \({\bf R}^4\):

\[ W = L((1,0,1,0), (2,0,0,0), (0,-3,-1,-1)) \]

and \(T\) given by the solution set of the system

\[ \begin{align*} x_1-x_2 & = 0 \\ x_1+x_2+x_3&= 0. \end{align*} \]

Determine \(T \cap W\).

Exercise 3.18

(See Solution 3.18.) Show that

\({\bf R}^2 = L((1,1), (2,-1))\),
\({\bf R}^2 = L((0,-2), (1,1))\).

Exercise 3.19

(See Solution 3.19.) Is \((1,5,0) \in {\bf R}^3\) a linear combination of \(v_1 = (1,1,0)\), \(v_2 = (2,0,1)\) and \(v_3 = (0,3,-1)\)?

(I.e., are there \(a_1, a_2, a_3 \in {\bf R}\) such that \(\alpha_1 v_1 + a_2 v_2 + a_3 v_3 = (1,5,0)\)?)

Exercise 3.20

(See Solution 3.20.) Express the following polynomials as \(f(x) = \sum_{i=0}^4 a_i (x-1)^i\):

\(f(x) = x^4\),
\(f(x) = x^3\),
\(f(x) = x^3 - 3x^2 + 4x+2\).

Exercise 3.21

(See Solution 3.21.) Let \(a, b \in {\bf R}\) be two distinct numbers. Show that the polynomials \(x-a\) and \(x-b\) are a basis of \({\bf R}[x]^{\le 1}\).

Exercise 3.22

(See Solution 3.22.) In \({\bf R}^4 = \{(x,y,z,t) \ | \ x,y,z,t \in {\bf R} \}\) consider the subspace \(W_1 \subset {\bf R}^4\) given by the solutions of the system

\[ \begin{align*} y+t&=0, \\ y+z&=0. \end{align*} \]

Also consider the subspace \(W_2 = L((0,1,-1,0))\).

Determine a basis and the dimension of \(W_1\). Describe \(W_1 \cap W_2\).

Exercise 3.23

(See Solution 3.23.) Let \(k \in {\bf R}\) be an arbitrary real number. Consider the subspace

\[ W_k := L((1,0,-1,0), (1,1,0,1), (1,2,k,1)) \subset {\bf R}^4. \]

For all \(k \in {\bf R}\), find a basis of \(W_k\) and determine \(\dim W_k\).
For which \(k \in {\bf R}\) is \((-1,1,1,1) \in W_k\)?

Exercise 3.24

(See Solution 3.24.) Recall that the dimension of the space \({\mathrm {Mat}}_{2 \times 3}\) of \(2 \times 3\)-matrices is 6.

Consider \(W = \left \{ \left ( \begin{array}{ccc} a & a+b & b \\ 0 & 0 & b \end{array} \right ) \ | \ a, b \in {\bf R} \right \}\). Confirm that \(W\) is a subspace of \({\mathrm {Mat}}_{2 \times 3}\). Determine \(\dim W\).
Let \(V := \left \{ \left ( \begin{array}{ccc} c & 0 & -c \\ 0 & 0 & -c \end{array} \right ) \ | \ c \in {\bf R} \right \}\). Determine (i.e., determine a basis and the dimension of) \(V \cap W\).

Exercise 3.25

(See Solution 3.25.) Consider the following subspaces of \({\bf R}^3\):

\[ \begin{align*} W_1 & := L((1,0,1), (2,1,0)) \\ W_2 & := L((-1,1,1), (0,3,0)). \end{align*} \]

Determine (i.e., determine a basis and the dimension of) \(W_1 \cap W_2\).
Determine \(W_1 + W_2\).

Exercise 3.26

(See Solution 3.26.) Consider the following subspaces of \({\bf R}^4\):

\[ \begin{align*} W_1 & := L((1,1,1,2), (2,0,3,5)) \\ W_2 & := L((1,1,0,1),(0,2,-2,-2)). \end{align*} \]

As in the previous exercise, determine \(W_1 \cap W_2\) and \(W_1 + W_2\).

Exercise 3.27

(See Solution 3.27.) Consider the subspace

\[ W = L(\underbrace{(1,0,1,0)}_{=v_1}, \underbrace{(2,0,1,1)}_{=v_2}, \underbrace{(0,0,1,3)}_{=v_3}). \]

Find a basis of \(W\) and determine \(\dim W\).
Find a vector \(v \in {\bf R}^4\) such that

\[ W \subsetneq L(v_1, v_2, v_3, v). \]

What is \(\dim L(v_1, v_2, v_3, v)\)?

Exercise 3.28

(See Solution 3.28.) Consider the vectors in \({\bf R}^4\), where \(t \in {\bf R}\):

\[ \begin{align*} u_1 &= (1,0,-1,2) \\ u_2 &=(1,0,0,1) \\ u_3&=(2,0,-1,3)\\ u_4&=(4,t,-2,6). \end{align*} \]

Let \(U_t = L(u_1, u_2, u_3, u_4)\) be the subspace spanned by these vectors (where the last vector depends on \(t \in {\bf R}\)). Find the values of \(t\) such that

\[ \dim U_t = 2. \]

Consider \(t = 1\) from now on. Verify \(\dim U_1 = 3\) and find a basis of \(U_1\).
Let \(W \subset {\bf R}^4\) be the subspace given by the equations

\[ \begin{align*} x_1+x_2+x_3 &= 0 \\ x_1 - 3x_4 &= 0. \end{align*} \]

Determine \(\dim W\) and \(\dim U_1 \cap W\).

Exercise 3.29

(See Solution 3.29.) Consider the subspace \(U_t \subset {\bf R}^4\) spanned by the four vectors

\[ \begin{align*} v_1 & = (1,0, 0, 1) \\ v_2 &= (-1,1,2,3) \\ v_3 &=(0,1,2,4) \\ v_4 &=(t,2,4,8). \end{align*} \]

Here, \(t \in {\bf R}\) is an arbitrary real number.

Find the values of \(t\), such that \(\dim U_t = 2\).
Consider from now on \(t = 1\). Determine \(\dim U_1\).
Let \(W \subset {\bf R}^4\) be the subspace given by the equations

\[ \begin{align*} x_1 - x_2 & =0\\ x_2 - x_3 & = 0. \end{align*} \]

Determine a basis and the dimension of \(W\) and of \(W \cap U_1\).