Question 1.12.4: Show that W is a subspace of R^4 and determine a basis for t...

First, we observe that a typical element v ofW is a vector of the form

v=\begin{bmatrix} 3a +b −c \\ 4a −5b +c \\ a +2b −3c \\ a −b \end{bmatrix}

Using properties of vector addition and scalar multiplication, we can write

v = a \begin{bmatrix} 3\\ 4\\ 1 \\ 1 \end{bmatrix} +b \begin{bmatrix} 1 \\ -5 \\ 2 \\ -1 \end{bmatrix} +c \begin{bmatrix} -1 \\ 1 \\ -3 \\ -1 \end{bmatrix}

From this, we observe that W may be viewed as the span of the set S = {w_{1},w_{2},w_{3}}, where

w_{1} =\begin{bmatrix} 3\\ 4\\ 1 \\ 1 \end{bmatrix} , w_{2} =\begin{bmatrix} 1 \\ -5 \\ 2 \\ -1 \end{bmatrix} , w_{3}= \begin{bmatrix} -1 \\ 1 \\ -3 \\ -1 \end{bmatrix}

As seen in exercise 19 in section 1.11, the span of any set of vectors in R^{n} generates a subspace of R^{n}; it follows thatW is a subspace of R^{4}. Moreover, we can observe that S = {w_{1},w_{2},w_{3}} is a linearly independent set since

\begin{bmatrix}3 & 1 &-1 \\ 4 & -5 &1 \\ 1 &2 &-3 \\ 1&-1&-1 \end{bmatrix} → \begin{bmatrix}1 & 0 &0 \\ 0 & 1 &0 \\ 0 &0 &1 \\ 0&0&0 \end{bmatrix}

Since S both spans the subspace W and is linearly independent, it follows that S is a basis for W.