For each of the items below, (i) If the kernel is non-trivial, find one non-zero
ID: 1259526 • Letter: F
Question
For each of the items below,
(i) If the kernel is non-trivial, find one non-zero element in it;
(ii) If the kernel is non-trivial, find two unequal non-zero vectors in the domain that are mapped by the linear transformation to the same vector in the range.
(a) The projection T : R3 to R3 onto the z-axis.
(b) The projection T : R2 to R2 onto the line defined by the equation y = 2x.
For each of the items below, (i) If the kernel is non-trivial, find one non-zero element in it; (ii) If the kernel is non-trivial, find two unequal non-zero vectors in the domain that are mapped by the linear transformation to the same vector in the range. (a) The projection T : R3 to R3 onto the z-axis. (b) The projection T : R2 to R2 onto the line defined by the equation y = 2x. (c) The matrix viewed as a linear transformation from R4 to R4 .Explanation / Answer
The Rank-Nullity Theorem tells you that if you have a linear transformation T:V?W, then the dimension of the kernel of T plus the dimension of the image of T equals the dimension of V. In this case, since your T maps R4 to R5, you know that the dimension of the kernel of T plus the dimension of Im(T) equals 4 (the dimension of R4). Since you know that dim(ker(T))?1
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