True or false khan sak2454 hw05 gilbert 57245 1 false

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Unformatted text preview: n TA : x → Ax is Rm . True or False? khan (sak2454) – HW05 – gilbert – (57245) 1. FALSE correct 005 2. TRUE Explanation: When A is m × n, the matrix product Ax is only defined for x in Rn , and the product is then a vector in Rm . Thus TA : x → Ax maps Rn to Rm . So the Range of TA is the set { TA (x) : x in Rn } of vectors in Rm . But this need not be all of Rm . For example, since 1 0 1 0 x1 x2 = x1 + x2 , 0 the range of TA does not contain any vector in R2 whose second entry is non-zero. Consequently, the statement is FALSE . 2 10.0 points If A is an m × n matrix, then the range of the transformation T : Rn → Rm , TA : x → A x , is the set of all linear combinations of the columns of A. True or False? 1. TRUE correct 2. FALSE Explanation: By definition, the range of TA : x → A x is the set {Ax : x in Rn } . But when A = [ a1 a2 . . . an ] , x1 x2 x = . , . . xn 004 10.0 points Every linear transformation T : Rn → Rm is a matrix transformation. True or False? 1. TRUE correct 2. FALSE Explanation: If T : Rn → Rm is a linear transformation, and A is t...
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