Science:Math Exam Resources/Courses/MATH307/April 2012/Question 03

MATH307 April 2012

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[hide] Question 03
Suppose that A is a 3 x 4 matrix and $\mathrm {rref} (A)={\begin{bmatrix}1&2&0&2\\0&0&1&2\\0&0&0&0\end{bmatrix}}$ For each of the following, either find what is asked for, or indicate that there is insufficient information to determine it: (i) the rank of A, (ii) a basis for N(A), (iii) a basis for R(A), (iv) a basis for N(AT), (v) a basis for R(AT).

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[show] Hint

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[show] Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. (i) The rank of A is determined by the number of pivots in rref(A). Therefore, the rank of A is 2. (ii) A basis for N(A) is determined by inspecting the free variables of rref(A), which are 2 and 4. Let $x_{2}=s$ and $x_{4}=t$. So, $x_{1}=-2x_{2}-2x_{4}$ and $x_{3}=-2x_{4}$. Then we can write: ${\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{bmatrix}}=s{\begin{bmatrix}-2\\1\\0\\0\end{bmatrix}}+t{\begin{bmatrix}-2\\0\\-2\\1\end{bmatrix}}$. Therefore, a basis for N(A) is ${\begin{bmatrix}-2\\1\\0\\0\end{bmatrix}}$ and ${\begin{bmatrix}-2\\0\\-2\\1\end{bmatrix}}$. (iii) A basis for R(A) are the columns of A corresponding to the pivot columns of rref(A). Since we don’t know A, we cannot determine a basis for R(A). (iv) A basis for N(AT) is determined by inspecting the free variables of rref(AT). Since we don’t know rref(AT), we cannot determine a basis for N(AT). (v) A basis for R(AT) are the rows of rref(A) that correspond to the pivots. Therefore a basis for R(AT) is ${\begin{bmatrix}1\\2\\0\\2\end{bmatrix}}$ and ${\begin{bmatrix}0\\0\\1\\2\end{bmatrix}}$.