Lecture16 - University of Toronto at Scarborough Department...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 16 Section 2.3 Sophie Chrysostomou 1 Linear Transformations theorem 1.1 If T : Rn −→ Rm is a linear transformation and v1 , v2 , · · · , vk ∈ Rn such that {T (v1 ), T (v2 ), · · · , T (vk )} is a linearly independent set in Rm . Then, {v1 , v2 , · · · , vk } is linearly independent also. proof. 1 theorem 1.2 (from Lecture 15) If T : Rn −→ Rm is a linear transformation, and B = {b1 , b2 , · · · , bn } a basis for Rn . Then: if v ∈ Rn =⇒ T (v) is determined by T (b1 ), T (b2 ), · · · , T (bn ). OR T [Rn ] ⊂ sp(T (b1 ), T (b2 ), · · · , T (bn ).) corollary 1.3 m × n matrix If T : Rn −→ Rm is a linear transformation, let A be the | | | A = T (e1 ) T (e2 ) · · · T (en ) | | | Then T (x) = Ax for all x ∈ Rn . A is called the standard matrix representation of T . (This is very powerful!!! It says that every linear transformation corresponds to a matrix multiplication.) 2 example 1.4 Suppose T : R4 −→ R3 is a linear transformation given by T ([x1 , x2 , x3 , x4 ]) = [x2 − 3x3 − x4 , 6x1 + 5x2 , x3 + 2x1 ]. Find the standard matrix representation of T . Homework: Suppose T : R3 −→ R5 is a linear transformation given by T ([x1 , x2 , x3 ]) = [2x1 + 3x3 − 4x2 , 3x1 − 2x3 , 7x2 + 4x3 , x1 − 4x2 + 2x3 , 5x2 ]. Find the standard matrix representation of T . Question: Let T : Rn −→ Rm be a linear transformation. How do we find the kernel of T ? How do we find the range of T ? Answer: Let A be the standard matrix representation of T . So T (x) = Ax for all x ∈ Rn . Remember, by definition: The nullspace or kernel of T is given by ker(T ) = T −1 [{0 }] = {v ∈ Rn T (v) = 0} = {v ∈ Rn Av = 0} = nullspace of A The range of T is given by T [Rn ] = {T (x) x ∈ Rn } = { Ax x ∈ R n } = column space of A 3 So: i) to find the nullspace (kernel) 0f T we find the nullspace of A. ii) to find the range of T , we find the column space of A. example 1.5 Suppose T : R4 −→ R3 is a linear transformation given by T ([x1 , x2 , x3 , x4 ]) = [x2 − 3x3 − x4 , 6x1 + 5x2 , x3 + 2x1 ]. Find a basis for the nullspace and a basis for the range of T . 4 definition 1.6 Let T : Rn −→ Rm be a linear transformation. • rank of T , denoted by rank (T ), is the dimension of the range of T . • nullity of T , denoted by nullity (T ), is the dimension of the kernel of T. • T is invertible if m = n and the standard matrix representation A of T is invertible. • the rank equation for linear transformations is: rank (T ) + nullity (T ) = dim(domain of T ) example 1.7 Let T : R4 −→ R4 be the linear transformation given by T ([x1 , x2 , x3 , x4 ]) = [x1 + x2 , x2 + x3 , −x4 + 2x2 + x3 , x1 + x4 ]. Find a basis for the range of T , a basis fo the nullspace of T , verify the rank equation, and find if T is invertible. 5 ...
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