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Category Archives: linear algebra
Linear Algebra and Its Applications, Exercise 3.1.12
Exercise 3.1.12. For the matrix find a basis for the nullspace and show that it is orthogonal to the row space. Take the vector and express it as the sum of a nullspace component and a row space component . … Continue reading
Posted in linear algebra
Tagged basis, nullspace, nullspace component, row space component
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Linear Algebra and Its Applications, Exercise 3.1.11
Exercise 3.1.11. Fredholm’s alternative to the fundamental theorem of linear algebra states that for any matrix and vector either 1) has a solution or 2) has a solution, but not both. Show that assuming both (1) and (2) have solutions … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.10
Exercise 3.1.10. Given the two vectors and find a homogeneous system in three unknowns whose solutions are the linear combinations of the vectors. Answer: In the previous exercise 3.1.9 we showed that the plane spanned by the vectors and was … Continue reading
Posted in linear algebra
Tagged homogeneous system, linear combinations, three unknowns
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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
Linear Algebra and Its Applications, Exercise 3.1.7
Exercise 3.1.7. For the matrix find vectors and such that is orthogonal to the row space of and is orthogonal to the column space of > Answer: The nullspace of is orthogonal to the row space of . We can … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.6
Exercise 3.1.6. What vectors are orthogonal to and in ? From these vectors create a set of three orthonormal vectors (mutually orthogonal with unit length). Answer: If is a vector orthogonal to both and then the inner product of with … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.5
Exercise 3.1.5. Of the following vectors which are orthogonal to one another? Answer: We have the following inner products among the vectors: So and are orthogonal to but not to each other. NOTE: This continues a series of posts containing … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.4
Exercise 3.1.4. If is an invertible matrix, describe why row of and column of are orthogonal in the case . Answer: We have . The identity matrix has ones on the diagonal (i.e., when ) and zeros otherwise (when ). … Continue reading
Linear Algebra and Its Applications, Exercise 3.1.3
Exercise 3.1.3. In the – plane two lines are perpendicular if the product of their slopes is -1. Use this fact to derive the condition for two vectors and being orthogonal. Answer: The line through the origin and has slope … Continue reading
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