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The problem is given below:

My result regarding to question 1 is given below(Calculated in maple):

Then, my question is how to solve the question 2? I'm in doubt what even to do. Could someone explain me, what to do?

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3 Answers

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The vectors that form a basis for the column space are certainly in the column space. You can choose one of these.

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Hint:

For the column space, pick any (nonzero) column.

For the row space, pick any (nonzero) row.

For the null space, notice that first and third columns of $A$ are equal, which means that

$$A\begin{bmatrix} 1 \\ 0 \\ -1 \\ 0\end{bmatrix} = 0.$$

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Other than just applying formulas, do you understand the meanings of these words? To get a vector in the column space, take any one of the columns of the matrix. To get a vector in the row space take any one of the rows of the matrix. A vector in the null space is any vector, v, such that Av= 0. That's the only non-trivial part of (2).

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