Exercise 3.5.1 In $\mathbb R^3$ let $\alpha_1=(1,0,1)$, $\alpha_2=(0,1,-2)$, $\alpha_3=(-1,-1,0)$. (a) If $f$ is a linear functional on $\mathbb R^3$ such that$$f(\alpha_1)=1,\quad f(\alpha_2)=-1,\quad f(\alpha_3)=3,$$and if $\alpha=(a,b,c)$, find $f(\alpha)$. (b) Describe explicitly…
Exercise 3.4.1 Let $T$ be the linear operator on $\mathbb C^2$ defined by $T(x_1,x_2)=(x_1,0)$. Let $\mathcal B$ be the standard ordered basis for $\mathbb C^2$ and let $\mathcal B'=\{\alpha_1,\alpha_2\}$ be…
Exercise 3.3.1 Let $V$ be the set of complex numbers and let $F$ be the field of real numbers. With the usual operations, $V$ is a vector space over $F$.…
Exercise 3.2.1 (a) Geometrically, in the $x-y$ plane, $T$ is the reflection about the diagonal $x=y$ and $U$ is a projection onto the $x$-axis. (b) We have $(U+T)(x_1,x_2)=(x_2,x_1)+(x_1,0)=(x_1+x_2,x_1)$. $(UT)(x_1,x_2)=U(x_2,x_1)=(x_2,0)$. $(TU)(x_1,x_2)=T(x_1,0)=(0,x_1)$.…
Exercise 3.1.1 (a) $T$ is not a linear transformation because $T(0,0)=(1,0)$ and according to the comments after Example 5 on page 68 we know that it must always be that…
Exercise 2.6.1 Let $\alpha_1$, $\alpha_2$, $\dots$, $\alpha_n$ be the colunms of $A$. Then $\alpha_i\in F^{s\times1}$ $\forall$ $i$. Thus $\{\alpha_1,\dots,\alpha_n\}$ are $n$ vectors in $F^{s\times1}$. But $F^{s\times1}$ has dimension $s<n$ thus…
Exercise 2.4.1 Using Theorem 7, page 52, if we calculate the inverse of $$P=\left[\begin{array}{cccc}1&0&1&0\\1&0&0&0\\0&1&0&0\\0&1&4&2\end{array}\right].$$then the columns of $P^{-1}$ will give the coefficients to write the standard basis vectors in terms…
Exercise 2.3.1 Suppose $v_1$ and $v_2$ are linearly dependent. If one of them, say $v_1$, is the zero vector then it is a scalar multiple of the other one $v_1=0\cdot…
Exercise 2.2.1 (a) This is not a subspace because for $(1,\dots,1)$ the additive inverse is $(-1,\dots,-1)$ which does not satisfy the condition. (b) Suppose $(a_1,a_2,a_3,\dots,a_n)$ and $(b_1,b_2,b_3,\dots,b_n)$ satisfy the condition…
Exercise 2.1.1 Example 1 starts with any field and defines the objects, the addition rule and the scalar multiplication rule. We must show the set of $n$-tuples satisfies the eight…