Math Solutions

(Solution Library) Consider the vectors in R^4 defined by: v_1=(-1,0,1,2), v_2=(3,4,-2,5), v_3=(1,4,0,9) Find the reduced row echelon form of the matrix

Question: Consider the vectors in \({{\mathbb{R}}^{4}}\) defined by:

\[{{\mathbf{v}}_{1}}=\left( -1,0,1,2 \right),\,\,{{\mathbf{v}}_{2}}=\left( 3,4,-2,5 \right),\,\,\,{{\mathbf{v}}_{3}}=\left( 1,4,0,9 \right)\]

Find the reduced row echelon form of the matrix which has these as its rows. What is its rank? Is \(\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) linearly independent? What is the dimension of \(span\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) ?

Find a homogeneous linear system for which the space of solutions is exactly the subspace of R4 spanned by v 1 , v 2 , and v 3 .

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