# Tag Archives: column space

## Linear Algebra and Its Applications, Exercise 3.4.15

Exercise 3.4.15. Given the matrix find the orthonormal vectors and that span the column space of . Next find the vector that completes the orthonormal set, and describe the subspace of of which is an element. Finally, for find the … Continue reading

## Linear Algebra and Its Applications, Exercise 3.4.1

Exercise 3.4.1. a) Given the following four data points: write down the four equations for fitting to the data. b) Find the line fit by least squares and calculate the error . c) Given the value of what is in … Continue reading

## Linear Algebra and Its Applications, Exercise 3.3.14

Exercise 3.3.14. Find the projection matrix onto the plane spanned by the vectors and . Find a nonzero vector that projects to zero. Answer: The plane in question is the column space of the matrix The projection matrix . We have … Continue reading

## Linear Algebra and Its Applications, Exercise 3.3.12

Exercise 3.3.12. Given the subspace spanned by the two vectors and find the following: a) a set of basis vectors for b) the matrix that projects onto c) the vector in that has the minimum distance to the vector in Answer: … Continue reading

## Linear Algebra and Its Applications, Exercise 3.3.8

Exercise 3.3.8. Suppose that is a projection matrix from onto a subspace with dimension . What is the column space of ? What is its rank? Answer: Suppose that is a arbitrary vector in . From the definition of we know … Continue reading

## Linear Algebra and Its Applications, Exercise 3.3.7

Exercise 3.3.7. Given the two vectors and find the projection matrix that projects onto the subspace spanned by and . Answer: The subspace spanned by  and is the column space where The projection matrix onto the subspace is then . We … Continue reading

## Linear Algebra and Its Applications, Exercise 3.3.6

Exercise 3.3.6. Given the matrix and vector defined as follows find the projection of onto the column space of . Decompose the vector into the sum of two orthogonal vectors and where is in the column space. Which subspace is in? … Continue reading

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## Linear Algebra and Its Applications, Exercise 3.3.5

Exercise 3.3.5. Given the system with no solution, provide a graph of a straight line that minimizes and solve for the equation of the line. What is the result of projecting the vector onto the column space of ? Answer: … Continue reading