ScienceMathAlgebraLinear Algebra 1.8 Concept Check
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Linear Algebra 1.8 Concept Check

Perla Roberts avatar Perla Roberts
7 September 2022
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10 test answers

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question
A linear transformation is a special type of function.
answer
True (A linear transformation is a function from R^n to ℝ^m that assigns to each vector x in R^n a vector ​T(x​) in ℝ^m)
question
If A is a 3×5 matrix and T is a transformation defined by ​T(x​)=Ax​, then the domain of T is ℝ3.
answer
False (The domain is actually ℝ^5​, because in the product Ax​, if A is an m×n matrix then x must be a vector in ℝ^n).
question
If A is an m×n ​matrix, then the range of the transformation x maps to↦Ax is ℝ^m.
answer
False (The range of the transformation is the set of all linear combinations of the columns of​ A, because each image of the transformation is of the form Ax.)
question
Every linear transformation is a matrix transformation.
answer
False (A matrix transformation is a special linear transformation of the form x to↦ Ax where A is a matrix.)
question
A transformation T is linear if and only if T(c1v1+c2v2)=c1T(v1)+c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2
answer
True (This equation correctly summarizes the properties necessary for a transformation to be linear.)
question
The range of the transformation x Ax is the set of all linear combinations of the columns of A.
answer
True (each image ​T(x​) is of the form Ax. ​Thus, the range is the set of all linear combinations of the columns of A.)
question
Every matrix transformation is a linear transformation.
answer
​True (every matrix transformation has the properties ​T(u+v​)=T(u​)+​T(v​) and ​T(cu​)=​cT(u​) for all u and v​, in the domain of T and all scalars c.)
question
If T: R^n ---) R^m is a linear transformation and if c is in R^m, then a uniqueness question is "Is c in the range of T?"
answer
​False( the question​ "is c in the range of​ T?" is the same as​ "does there exist an x whose image is c​?" This is an existence question.)
question
A linear transformation preserves the operations of vector addition and scalar multiplication.
answer
True; The linear transformation ​T(cu+dv​) is the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are preserved.
question
A linear transformation T: R^n ---) R^m always maps the origin of R^n to the origin of R^m
answer
​True (for a linear​ transformation, ​T(0​) is equal to 0.)