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farkhsLv1
11 Mar 2023
Q1. Let 
, 
be non-zero orthogonal vectors and let T: 
be the linear transformation defined by
T(
) = 2 
() + 2 
() -
and let A = [T] 
be its standard matrix.
(a) Prove that 
. [Hint: Do not attempt to "compute" A. Instead, think about how 
is related to T.]
(b) Using (a), determine whether T is one-to-one and/or onto. Justify your answers.
(c) Consider the case where
and 
. Determine 
explicitly as a vector in . Hence, determine A in this particular case.
(d) Returning now to to the general case, let 
= 
{, } be the plane in spanned by and . Referring to , give a one-sentence description of what T does geometrically. No justification is necessary. [Hint: Use part (c) for inspiration. A correct description will allow you to immediately see why is true.]
Q1. Let , be non-zero orthogonal vectors and let T: be the linear transformation defined by
, be non-zero orthogonal vectors and let T: be the linear transformation defined byT() = 2 () + 2 () -
) = 2 () + 2 () - and let A = [T] be its standard matrix.
be its standard matrix.(a) Prove that . [Hint: Do not attempt to "compute" A. Instead, think about how is related to T.]
. [Hint: Do not attempt to "compute" A. Instead, think about how is related to T.](b) Using (a), determine whether T is one-to-one and/or onto. Justify your answers.
(c) Consider the case where and . Determine explicitly as a vector in . Hence, determine A in this particular case.
and . Determine explicitly as a vector in . Hence, determine A in this particular case.(d) Returning now to to the general case, let = {, } be the plane in spanned by and . Referring to , give a one-sentence description of what T does geometrically. No justification is necessary. [Hint: Use part (c) for inspiration. A correct description will allow you to immediately see why is true.]
= {, } be the plane in spanned by and . Referring to , give a one-sentence description of what T does geometrically. No justification is necessary. [Hint: Use part (c) for inspiration. A correct description will allow you to immediately see why is true.]
12 Mar 2023
heyshivanshuLv6
12 Mar 2023
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); decreasing: (-
in the x-axis. What is a function rule for g(x) ?
. The graph of g(x) is f(x) translated to the left 6 units and down 5 units. What is the function rule for g(x)?
