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Chapter 2: Q2.8-10Q (page 93)

10: With \({\mathop{\rm u}\nolimits} = \left( { - 2,3,1} \right)\) and A as in Exercise 8, determine if u is in Nul A.a

Short Answer

Expert verified

\({\mathop{\rm u}\nolimits} \)is in Nul A.

Step by step solution

01

State the value of A as in Exercise 8

Matrix A in the form \(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}\end{array}} \right]\), as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 2}&0\\0&2&{ - 6}\\6&3&3\end{array}} \right]\)

02

Determine whether p is in Nul A

The null spaceof matrix A is the set Nul Aof all solutions of the homogeneous equation\(Ax = 0\).

Calculate \(A{\mathop{\rm u}\nolimits} \), as shown below:

\(\begin{array}{c}A{\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 2}&0\\0&2&{ - 6}\\6&3&3\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\\1\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{6 - 6 + 0}\\{0 + 6 - 6}\\{ - 12 + 9 + 3}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

Since\[A{\mathop{\rm u}\nolimits} = 0\], \({\mathop{\rm u}\nolimits} \) is in Nul A.

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Most popular questions from this chapter

[M] For block operations, it may be necessary to access or enter submatrices of a large matrix. Describe the functions or commands of your matrix program that accomplish the following tasks. Suppose A is a \(20 \times 30\) matrix.

  1. Display the submatrix of Afrom rows 15 to 20 and columns 5 to 10.
  2. Insert a \(5 \times 10\) matrix B into A, beginning at row 10 and column 20.
  3. Create a \(50 \times 50\) matrix of the form \(B = \left[ {\begin{array}{*{20}{c}}A&0\\0&{{A^T}}\end{array}} \right]\).

[Note: It may not be necessary to specify the zero blocks in B.]

Let \(S = \left( {\begin{aligned}{*{20}{c}}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{aligned}} \right)\). Compute \({S^k}\) for \(k = {\bf{2}},...,{\bf{6}}\).

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. [Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)].

Show that if ABis invertible, so is B.

Let \(A = \left[ {\begin{array}{*{20}{c}}B&{\bf{0}}\\{\bf{0}}&C\end{array}} \right]\), where \(B\) and \(C\) are square. Show that \(A\)is invertible if an only if both \(B\) and \(C\) are invertible.
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