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Tag Archives: orthogonal subspaces
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
Posted in linear algebra
Tagged column space, left nullspace, orthogonal subspaces, orthogonal vectors, projection matrix
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Linear Algebra and Its Applications, Exercise 3.1.20
Exercise 3.1.20. Suppose is a subspace of . Show that . What does this mean? Answer: We first consider the case where ; in other words, contains only the zero vector. From exercise 3.1.18 we know that . The only … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.19
Exercise 3.1.19. State whether each of the following is true or false: (a) If the subspaces and are orthogonal, then and are also orthogonal. (b) If is orthogonal to and orthogonal to then is orthogonal to . Answer: (a) In … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.9
Exercise 3.1.9. For the plane in spanned by the vectors and find the orthogonal complement (i.e., the line in perpendicular to the plane). Note that this can be done by solving the system where the two vectors are the rows … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.8
Exercise 3.1.8. Suppose that and are orthogonal subspaces. Show that their intersection consists only of the zero vector. Answer: If and are orthogonal then we have for any vectors in and in . Suppose that is an element of both … Continue reading