0

Linear Algebra: linear algebra – #28064

Question: Recall that we use the symbol \(\mathbb{R}\left[ x \right]\) to mean the real vector space of all polynomials in x with real coefficients. Show that the set \[\left\{ p\left( x \right)\in \mathbb{R}\left[ x \right]:\,\,p\left( 1 \right)=0\text{ and }\deg \left( p\left( x \right) \right)\le 3 \right\}\] is a subspace of \(\mathbb{R}\left[ x \right]\). Is it...

0

Linear Algebra: linear algebra – #28063

Question: Express \(2{{x}^{3}}-9{{x}^{2}}+16x-9\) as a linear combination of \(x-1\), \({{\left( x-1 \right)}^{2}}\), and \({{\left( x-1 \right)}^{3}}\). That is, find real numbers \({{k}_{1}},{{k}_{2}},{{k}_{3}}\) so that \[2{{x}^{3}}-9{{x}^{2}}+16x-9={{k}_{1}}\left( x-1 \right)+{{k}_{2}}{{\left( x-1 \right)}^{2}}+{{k}_{3}}{{\left( x-1 \right)}^{3}}\]

0

Linear Algebra: linear algebra – #28051

Question: Consider the vectors in \({{\mathbb{R}}^{4}}\) defined by: \[{{\mathbf{v}}_{1}}=\left( -1,0,1,2 \right),\,\,{{\mathbf{v}}_{2}}=\left( 3,4,-2,5 \right),\,\,\,{{\mathbf{v}}_{3}}=\left( 1,4,0,9 \right)\] Find the reduced row echelon form of the matrix which has these as its rows. What is its rank? Is \(\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) linearly independent? What is the dimension of \(span\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) ? Find a homogeneous linear system for...

log in

reset password

Back to
log in