Apologize, completely understood the question now.

Given: \begin{align*} v_1 &= (1, −1, 2, 0) \\ v_2 &= (1, 0, 1, 1) \\ v_3 &= (1, −2, 3, −1) \\ v_4 &= (3, 1, 2, 4) \end{align*}

Are $v_1, v_2, v_3, v_4$ linearly independent? I was confused with the matrix linearly independent.

How is vectors linearly independently vs matrix linearly independently?

  • $\begingroup$ Check your arithmetic. $\endgroup$ – amd Jan 9 at 8:01
  • $\begingroup$ So you have found that only three of the row vectors are linearly independent. Why do you think the rank is 2? $\endgroup$ – almagest Jan 9 at 8:24
  • $\begingroup$ the rank is 2 because only R1 and R2 are the ones considered. since 1 in the leading position of R1 and R2. $\endgroup$ – user740188 Jan 9 at 8:29
  • $\begingroup$ You have misunderstood the meaning of rank. What matters is whether the vectors are linearly independent. The rank is defined as the dimension of the vector space spanned by the row vectors. The vector space spanned by (1,-1,2,0),(0,1,-1,1) and (0,0,-4,4) is the same as the vector space spanned by (1,-1,2,0),(0,1,-1,1) and (0,0,1,-1). $\endgroup$ – almagest Jan 9 at 8:39
  • $\begingroup$ @almagest, I tried to understand "The rank is defined as the dimension of the vector space spanned by the row vectors." but seems like even more confusing to me. Can you please give an example? $\endgroup$ – user740188 Jan 9 at 9:23

If that is the REF that you obtained, you could have divide the third row by $-4$ and you will get a leading one in that row and the conclusion is the rank is $3$.

However, that is not the right REF. Check the step when you first update $R_3$, you make an arithmetic mistake there. The rank is indeed $2$.

Also notice that $cR_i-R_j$ is not actually a single elementary operations, it consist of $-R_i + R_j$ and $-R_j$.

If you have a zero row in the end, clearly they are not linearly indepedent.

If your original matrix is $A$, working on $A^T$ will help you identify a basis in the original vector set. Also, note that since there is no row swapping operations, the first two rows are linearly indepedent.

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  • $\begingroup$ What do you mean by REF? "the first two rows are linearly independent?" The solutions given v1,v2, v3, v4 are linearly dependent. $\endgroup$ – user740188 Jan 9 at 9:40
  • $\begingroup$ REF means row echelon form. $\{v_1, v_2\}$ are linearly independent, $\{v_1, v_2, v_3, v_4\}$ are linearly dependent. $\endgroup$ – Siong Thye Goh Jan 9 at 10:02
  • $\begingroup$ Apologize, new to matrix, can't even understand the question correctly, after check the solution provided then realized my misunderstood the question. $\endgroup$ – user740188 Jan 9 at 10:08

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