For a line L in ℝ 2 , draw a sketch to interpret the following transformations geometrically: a. T ( x → ) = x → − proj L x → b. T ( x → ) = x → − 2 proj L x → c. T ( x → ) = 2 proj L x → − x →
For a line L in ℝ 2 , draw a sketch to interpret the following transformations geometrically: a. T ( x → ) = x → − proj L x → b. T ( x → ) = x → − 2 proj L x → c. T ( x → ) = 2 proj L x → − x →
Solution Summary: The author illustrates the transformation of T(stackrelto x) into a component proj_L
For a line L in
ℝ
2
, draw a sketch to interpret the following transformations geometrically: a.
T
(
x
→
)
=
x
→
−
proj
L
x
→
b.
T
(
x
→
)
=
x
→
−
2
proj
L
x
→
c.
T
(
x
→
)
=
2
proj
L
x
→
−
x
→
Let T be a linear transformation from R2 to R2 (or from R3 to R3). Prove that T maps a straight line to a straight line or a point.
For Exercises 2 through 6, prove that T is a linear transformation, and find bases for both N(T) and R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto.
5. T: P2(R) → P3(R) defined by T(f(x)) = xf(x) + f’(x).
Let L : Mnm → Mmn be the function defined by L (A) = AT (the transpose of A), for A in V .Is L a linear transformation? Justify your answer.
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