xiii Note to the reader: This is a customized book for Old Dominion ...
xiii Note to the reader: This is a customized book for Old Dominion ...
xiii Note to the reader: This is a customized book for Old Dominion ...
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What do you mean by <strong>the</strong> rank of a set of vec<strong>to</strong>rs?Prove that if a set of vec<strong>to</strong>rs contains <strong>the</strong> null vec<strong>to</strong>r, <strong>the</strong> set of vec<strong>to</strong>rs <strong>is</strong> linearlydependent.Prove that if a set of vec<strong>to</strong>rs are linearly independent, <strong>the</strong>n a subset of <strong>the</strong> m vec<strong>to</strong>rs alsohas <strong>to</strong> be linearly independent.Prove that if a set of vec<strong>to</strong>rs <strong>is</strong> linearly dependent, <strong>the</strong>n at least one of <strong>the</strong> vec<strong>to</strong>rs can bewritten as a linear combination of o<strong>the</strong>rs.Prove that if <strong>the</strong> dimension of a set of vec<strong>to</strong>rs <strong>is</strong> less than <strong>the</strong> number of vec<strong>to</strong>rs in <strong>the</strong> set,<strong>the</strong>n <strong>the</strong> set of vec<strong>to</strong>rs <strong>is</strong> linearly dependentHow can vec<strong>to</strong>rs be used <strong>to</strong> write simultaneous linear equations?What <strong>is</strong> <strong>the</strong> definition of <strong>the</strong> dot product of two vec<strong>to</strong>rs?Chapter 04.03 Binary matrix operationsHow do you add two matrices?How do you subtract two matrices?How do I multiply two matrices?What <strong>is</strong> a scalar product of a constant and a matrix?What <strong>is</strong> a linear combination of matrices?What are some of <strong>the</strong> rules of binary matrix operations?Commutative law of additionAssociative law of additionAssociative law of multiplicationD<strong>is</strong>tributive lawIs [A][B] = [B][A]?Chapter 04.04 Unary matrix operationsWhat <strong>is</strong> <strong>the</strong> transpose of a matrix?What <strong>is</strong> a symmetric matrix?What <strong>is</strong> a skew-symmetric matrix?What <strong>is</strong> <strong>the</strong> trace of a matrix?Define <strong>the</strong> determinant of a matrix.Is <strong>the</strong>re a relationship between det(AB), and det(A) and det(B)?Are <strong>the</strong>re some o<strong>the</strong>r <strong>the</strong>orems that are important in finding <strong>the</strong> determinant?Chapter 04.05 System of equationsMatrix algebra <strong>is</strong> used <strong>for</strong> solving systems of equations. Can you illustrate th<strong>is</strong> concept?A system of equations can be cons<strong>is</strong>tent or incons<strong>is</strong>tent. What does that mean?How can one d<strong>is</strong>tingu<strong>is</strong>h between a cons<strong>is</strong>tent and incons<strong>is</strong>tent system of equations?If a solution ex<strong>is</strong>ts, how do we know whe<strong>the</strong>r it <strong>is</strong> unique?If we have more equations than unknowns in [A] [X] = [C], does it mean <strong>the</strong> system <strong>is</strong>incons<strong>is</strong>tent?Can I use <strong>the</strong> concept of <strong>the</strong> inverse of a matrix <strong>to</strong> find <strong>the</strong> solution of a set of equations[A] [X] = [C]?How do I find <strong>the</strong> inverse of a matrix?Is <strong>the</strong>re ano<strong>the</strong>r way <strong>to</strong> find <strong>the</strong> inverse of a matrix?If <strong>the</strong> inverse of a square matrix [A] ex<strong>is</strong>ts, <strong>is</strong> it unique?Chapter 04.06 Gaussian elimination 220How are a set of equations solved numerically? 220Forward elimination of unknowns 220Back substitution 222xvii
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