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An affine transformation \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) has the form \({\bf{T}}\left( {\bf{x}} \right) = {\bf{A}}\), with \({\bf{A}}\) an \({\bf{m}} \times {\bf{n}}\) matrix and \({\bf{b}}\) in \({\mathbb{R}^{\bf{m}}}\). Show that \({\bf{T}}\) is not a linear transformation when \({\bf{b}} \ne {\bf{0}}\). ( Affine transformations are important in computer graphics. )

a. Fill in the blank in the following statement: 鈥淚f \(A\) is a \(m \times n\) matrix, then the columns of \(A\) are linearly independent if and only if \(A\) has _______ pivot columns.鈥�

b. Explain why the statement in (a) is true

In Exercises 29 鈥� 32, (a) does the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) have a nontrivial solution and (b) does the equation \(Ax = b\) have at least one solution for every possible \({\mathop{\rm b}\nolimits} \)?

30. \(A\) is a \(3 \times 3\) matrix with three pivot positions.

Question: In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation\(T\). Use the notation of Example 1 in section 1.2.

30. \(T:{\mathbb{R}^4} \to {\mathbb{R}^3}\) is onto.

Construct a \(3 \times 3\) matrix, not in echelon form, whose columns do not span \({\mathbb{R}^3}\). Show that the matrix you construct has the desired property.

Give an example of an inconsistent underdetermined system of two equations in three unknowns.

Let \(v\) be the center of mass of a system of point masses located at \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}\) as in exercise 29. Is \(v\) in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_k}} \right\}\)? Explain.

If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

Find the polynomial of degree 2[a polynomial of the form $\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{bt}\mathbf{+}{\mathbf{ct}}^{\mathbf{2}}$] whose graph goes through the points localid="1659342678677" $\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{}\mathbf{and}\mathbf{}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{13}\mathbf{)}\mathbf{.}\mathbf{}$Sketch the graph of the polynomial.

Let A be a \(3 \times 2\) matrix. Explain why the equation \(A{\bf{x}} = {\bf{b}}\) cannot be consistent for all b in \({\mathbb{R}^3}\). Generalize your argument to the case of an arbitrary A with more rows than columns.