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Chapter 2: Q2.8-30Q (page 93)

Suppose the columns of a matrix \(A = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm a}\nolimits} _1}}& \ldots &{{{\mathop{\rm a}\nolimits} _p}}\end{array}} \right]\) are linearly independent. Explain why \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\) is a basis for Col A.

Short Answer

Expert verified

The set \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\) is a basis for Col A.

Step by step solution

01

Condition for the basis of Col A

Thepivot columnsof matrix A forms a basisfor the column space of A.

02

Explain \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\)as a basis for Col A

Abasisfor a subspace H of \({\mathbb{R}^n}\) is a linearly independentset in H that spans H.

The set \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\) spans Col A because Col A is the set of all linear combinations of \({{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}\). The set \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\) is also linearly independent. Hence, it is a basis for Col A.

Thus, the set \(\left\{ {{{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _p}} \right\}\) is a basis for Col A.

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Most popular questions from this chapter

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.

If a matrix \(A\) is \({\bf{5}} \times {\bf{3}}\) and the product \(AB\)is \({\bf{5}} \times {\bf{7}}\), what is the size of \(B\)?

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?

Explain why the columns of an \(n \times n\) matrix Aare linearly independent when Ais invertible.
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