How to check if the columns of a given vector spans Rn

Question

Q1: In this question, find out if the given vectors $\{v_1,v_2,v_3\}$ span $R^4$. Vectors $v_1,v_2,v_3$

Q2: Given this matrix Matrix $B$ . Find out if the columns of this matrix span $R^4$.

I came across several solutions for these two questions. In all of those augmented matrix was made and checked for pivot columns. My question is why are we creating augmented matrix to check the span ? We should rather be making an equation like $[A]X = b$, where $A$ is the given matrix in the question, and then check for consistency.

Answer 1

Remember that one way to solve the equation $Ax = b$ is to make an augmented matrix, and we know that the system is consistent if it has the right amount of pivot columns. So what you saw online is probably just people doing exactly what you were describing.

One thing to remember is that 3 vectors can't span $\mathbb{R}^4$ you need at least 4 vectors, so for the first question, we don't even need to look at the vectors to know that we won't have enough vectors to span all of $\mathbb{R}^4$

For your second question, to see if the columns of the matrix span $\mathbb{R}^4$, all we need to do is row reduce the matrix. If we get the identity, then we'll span $\mathbb{R}^4$, and if we don't, (i.e. we get a row of all zeros somewhere) then that means there's some inconsistency in the system $Ax = b$ for some vector $b$ in $\mathbb{R}^4$ Thus the columns don't span all of $\mathbb{R}^4$.

when I row reduced I got the matrix $\begin{bmatrix} 1&0&-5&0\\ 0&1&1&0\\ 0&0&0&1\\ 0&0&0&0 \end{bmatrix}$

thus, the columns don't span $\mathbb{R}^4$, (I only have 3 pivot positions not 4, and I got a row of all $0$'s) I hope that helps! Good luck studying!

Answer 2

Let say we have a set $\{v_1,v_2,v_3\}$ a vector $v_4\in Span(\{v_1,v_2,v_3\})$ iff $v_4=\alpha v_1+\beta v_2+\gamma v_3$ for some $\alpha ,\beta,\gamma$ in a field.

Therefore the system of linear equations that are created is of the form $Ax=b$ where the matrix $A$ is the vectors as columns ($\{v_1,v_2,v_3\}$), the vector $x$ is the coefficients ($\alpha ,\beta,\gamma$) in the case above) and the vector $b$ is the vector we want to check if he in the span ($v_4$)

If the augmented matrix is inconsistent so there are no coefficients ($\alpha ,\beta,\gamma$) such that $v_4\in Span(\{v_1,v_2,v_3\})$ If it is consistent there are ($\alpha ,\beta,\gamma$) such that $v_4\in Span(\{v_1,v_2,v_3\})$