Question

3, (6 points) Let T : R4 ? R4 be linear and A be its standard matrix (T(x) = Ax). If dim range(T) 3, explain why one of the columns of A must be a linear combination of the other columns. 4. (6 points) Let a - 2b a. be a subset of R3. Is W a subspace of R3? Justify your answer.

These are two parts of single question. I uploaded it twice but got wrong answer though both the parts are very simple. Hope to get right answer asap thanks

Answer 1

1) Given T : R⁴ $\to$ R⁴ is linear and A is its standard matrix ,i.e., T(x) = Ax.

Also given that range(T) = 3.

Now, range(T) = 3 implies that there are 3 linearly independent columns.

And, T : R⁴ $\to$ R⁴ implies that the standard matrix A has 4 columns.

Then, range(T) = 3 and T : R⁴ $\to$ R⁴ together imply that there are 3 linearly independent columns and the other column is linear combination of these 3 linearly columns.

2) Given W = { $\begin{bmatrix} a-2b\\ b\\ a \end{bmatrix}$ : a,b $\in$ R } is a subset of R³.

Let A = $\begin{bmatrix} a-2b\\ b\\ a \end{bmatrix}$ $\in$ W and B = $\begin{bmatrix} x-2y\\ y\\ x \end{bmatrix}$ $\in$ W.

Let c,d $\in$ F.

Now, cA+dB = c $\begin{bmatrix} a-2b\\ b\\ a \end{bmatrix}$ +d $\begin{bmatrix} x-2y\\ y\\ x \end{bmatrix}$ = $\begin{bmatrix} (ca+dx)-2(cb+dy)\\ cb+dy\\ ca+dx \end{bmatrix}$ $\in$ W.

Then, for all A,B $\in$ W we have cA+dB $\in$ W.

Therefore, W is a subspace of R³.