For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of AA=[235−9−8−9−11214−3−1727]&nbsp;Find a vector in the null space of A that is not the zero vector&nbsp;A=[−3201]

Question

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of AA=[235−9−8−9−11214−3−1727]&amp;nbsp;Find a vector in the null space of A that is not the zero vector&amp;nbsp;A=[−3201]

Willie · Accepted Answer

Examine the following matrix,&amp;nbsp;A=[235−9−8−9−11214−3−1727]&amp;nbsp;Find a vector that is not in the null space of AUse the reduced row methodA=[13/25/2−9/2−8−9−11214−3−1727]&amp;nbsp;R1→12R1&amp;nbsp;≈[13/25/2−9/2039−150−9−2745]&amp;nbsp;R2→R2+8R1,R3→R3−4R1&amp;nbsp;≈[13/25/2−9/2039−150000]&amp;nbsp;R3→R3+3R2&amp;nbsp;≈[10−23013−50000]&amp;nbsp;R1→R1−12R2&amp;nbsp;The matrix has 2 pivot columns and 2 free columns.&amp;nbsp;Then, AX=0&amp;nbsp;[10−23013−50000][x1x2x3x4]=[000]&amp;nbsp;x1−2x3+3x4=0⇒x1=2x3−3x4&amp;nbsp;x2+3x3−5x4=0⇒x2=−3x3+5x4&amp;nbsp;So, the general solution is,&amp;nbsp;{x1=2x3−3x4x2=−3x3+5x4x3&amp;nbsp;and&amp;nbsp;x4&amp;nbsp;are free variables&amp;nbsp;&amp;nbsp;Then, a vector parametric solution is,&amp;nbsp;[x1x2x3x4]=[2x3−3x4−3x3+5x4x3x4]&amp;nbsp;=[2x3−3x3x3x4]+[−3x45x40x4]&amp;nbsp;=[2−310]x3+[−3501]x4&amp;nbsp;Therefore, the required non-zero vectors in Nul A are,&amp;nbsp;(2,−3,1,0),(−3,5,0,1)&amp;nbsp;Find a non zero vector in Col A&amp;nbsp;observe the equation (1), the first two columns of the matrix&amp;nbsp;&amp;nbsp;

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. A=\begin{bmatrix}2&3&5&-9\\-8&-9&-11&21\\4&-3&-17&27\end{bmatrix} Find a nonzero vector in Nul A. A=\begin{bmatrix}-3\\2\\0\\1\end{bmatrix}

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