3.1 Day 2 Applications and properties of a Kernel.

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Presentation transcript:

3.1 Day 2 Applications and properties of a Kernel

3.1 day 2 Applications and properties of a Kernel …Wait… No, wrong slide… not that kind of a kernel…

Properties of invertible matrices The following statements are equivalent for an nxn matrix 1 A is invertible 2 The linear system Ax = b has a unique solution x for all b 3rref(A) = I 4Im(A) = R n 5Rank (A) = n 6Ker(A) = 0

Properties of an image of a Linear Transformation 1.The zero vector is in the image of T 2. The image of T is closed under addition. (if two vectors are in the image of T then the sum of those two vectors is also in the image of T) 3. The image of T is closed under scalar multiplication. if a vector is in the image then any scalar multiple of that vector is also in the image)

Prove Ker(A)= 0 for all invertible matrices

Problem 22 Find the image by both inspection and rref

Solution to Problem 22 To see if the last column is a scalar multiple of the first two use 2x +y =3 x =2 y = -1 3x+4y = 2 Find the values of x and y and see if they work in the equation 6x+5y = 7

Example 7

Example: 7 Solution

Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25

Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25 (Hint this is a projection matrix onto

Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25 This matrix is a projection matrix onto the line that consists of the scalar multiples of The image is the line (the scalar multiples of ) The kernel are vectors that are perpendicular to the given line.

Find the image and kernel of the following matrix -3/5 3/5 4/5 4/5 Use your library of basic matrices to obtain the answer

Find the image and kernel of the following matrix -3/5 3/5 4/5 4/5 This is a rotation matrix the image is R 2 The Kernel is the zero vector

Problem 24 Describe the Image and the Kernel of the transformation

Problem 24 Solution

Problem 37

37 Solution

Problem 38

38 Solution

Homework p ,25,27,29,39,41,44,49

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