Search results

  1. S

    Basis of eigenvectors

    Right the basis for the kernel is the span of ##(1,1,1)##. Yes eigenvectors are linearly independent so they do span the range thanks!
  2. S

    Basis of eigenvectors

    So the basis for the range of ##T## are the other two eigenvectors.
  3. S

    Basis of eigenvectors

    Ok so the kernel of ##T## is ##(x,y,z)## such that ##T(x,y,z)=0## & this only occurs when we have ## (1,1,1)## so I guess that is the basis for the kernel right???
  4. S

    Basis of eigenvectors

    Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!!!
  5. S

    Showing a matrix A is diagonalisable

    Show that ##\{\mathbf{v}_1\}## is linearly independent. Simple enough lets consider $$c_1\mathbf{v}_1 = \mathbf{0}.$$ Our goal is to show that ##c_1 = 0##. By the definition of eigenvalues and eigenvectors we have ##A\mathbf{v}_1= \lambda_1\mathbf{v}_1##. Lets multiply the above equation and...
  6. S

    Finding ##(g\circ f)'(6)##

    The solution is 3: It's just ##(g\circ f)'(6) = (-1,-2)\cdot (4,-3) = (-1\times 4)+((-2)\times (-3)) = -4+6 = 2## using the multi-variate chain rule and the dot product. Is this correct and if not how do I go about doing it? Thanks!
  7. S

    Multivariate calculus problem: Calculating the gradient vector

    Okay then I’m lost how do we then justify whether to use the chain rule???
  8. S

    Multivariate calculus problem: Calculating the gradient vector

    Okay (1) and (2) are done. So for (3), assuming ##t > 0##, ##f\circ \mathbf{r} = \ln{(e^{\sin{(t)}})}^2+\ln{(e^{\sin{(t)}})}^2 = \sin^2{(t)}+\cos^2{(t)} = 1## so the derivative is ##0##.
  9. S

    Prove that the set T:={x∈Rn:Ax∈S} is a subspace of Rn.

    Yeah that certainly doesn't make sense!!! 1. Suppose ##\mathbf{0}\in T##, ##A(\mathbf{0})\in S## which is non-empty. 2. Suppose ##x_1,x_2\in T##. We then have that ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##, i.e. ##x_1+x_2\in T##. 3. Suppose ##\mathbf{x}\in T##, with ##\lambda\in \mathbb{R}##...
  10. S

    Prove that the set T:={x∈Rn:Ax∈S} is a subspace of Rn.

    Thanks!!! 1.Since ##\mathbf{0}\in T##, ##A(\mathbf{0})\in S## which is non-empty. 2. Suppose ##x_1,x_2\in T##. Then there exists vectors ##x_1,x_2\in S## such that we have ##A(x_1)## and ##A(x_2)##. We then have that ##x_1+x_2\in S## and ##A(x_1+x_2) = A(x_1)+A(x_2)##, i.e. ##x_1+x_2\in T##. 3...
  11. S

    Multivariate calculus problem: Calculating the gradient vector

    The derivative of ##\mathbf{r}## at each point of ##(0,1)##???
  12. S

    Multivariate calculus problem: Calculating the gradient vector

    1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...
  13. S

    Prove that the set T:={x∈Rn:Ax∈S} is a subspace of Rn.

    1. Lets show the three conditions for a subspace are satisfied: Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##. Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##. Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) =...
  14. S

    Multivariate problem

    There are sets of the form ##\left\{(x,y)\in \mathbb{R}^2: f(x,y) = \ln{\left(3+(x+y)^2\right)} = c\right\}## where ##c## is some fixed number ##> 1##. Lets see what happens for a few values of ##c##. Suppose ##c = 2##, then ##\ln{\left(3+(x+y)^2\right)} = 2 \Longleftrightarrow (x+y)^2 =...
  15. S

    Area between 2 curves

    Hold up. I can just calculate ##\int_{0.5}^{1} (2x-1)^4 \; dx=0.1##. Then calculate ##\int_{7/8}^{1} 8x-7 \; dx=0.0625##. Now take the difference to get ##\frac{3}{80}=0.0375##.
  16. S

    Area between 2 curves

    So that means I solve ##\int_{0.5}^{1} (2x-1)^4 -8x + 7 \; dx## in which case I get ##0.6##???
  17. S

    Area between 2 curves

    Oops no it isn’t that will be ##x=1##.
  18. S

    Area between 2 curves

    I’m thinking it’s where the tangent function cuts the ##x##-axis which is when ##8x-7=0##, or ##x=\frac{7}{8}##.
  19. S

    Area between 2 curves

    Yes it’s ##x=0.5##.
  20. S

    Minimum length of wire

    1. Homework Statement A wire pattern is inserted into a ##10##cm square by making a horizontal line in the middle of the square (not all the way across and with length ##x##) and connecting the ends of this line to the closest two corners. What is the minimum value of ##x##? 2. Homework...
  21. S

    Area between 2 curves

    Ok I found the equation of the tangent curve in the standard way to get ##y=8x-7## which cuts the x axis at ##x=\frac{7}{8}##. To find P I found the x coordinate Of the turning point which is ##x=0.5## so now we have our bounds of our integral so we just calculate ##\int_{0.5}^{7/8}...
  22. S

    Area between 2 curves

    I seem to read it fine once I open the attachment. It is asking given we have ##f(x) = (2x-1)^4##. The curve meets the ##x##-axis at a point ##P## and the line on the graph is a tangent to the curve at the point ##Q(1,1)##. Find the area of the region bounded by the curve, the ##x##-axis, and...
  23. S

    Area between 2 curves

    Member warned to type the problem statement, not just post an image with type that is too small to read 1. Homework Statement See attached. 2. Homework Equations 3. The Attempt at a Solution Ok so the first thing you wanna do is find the equation of the tangent line which is done in the...
  24. S

    Parametric equation

    Got it thanks a lot
  25. S

    Parametric equation

    1. Homework Statement A curve is defined by the parametric equations ##x=t^3+1## and ##y=t^2+1##. Show that ##\frac{\frac{d^2y}{dx^2}}{\left(\frac{dy}{dx}\right)^4}## is a constant. 2. Homework Equations 3. The Attempt at a Solution So you differentiate both equations wrt ##t## then apply...
  26. S

    Complex number problem: If arg(w)=π4 and |w⋅¯w|=20, then what is w of the form a+bi

    I do. ##w=-\sqrt{10}-\sqrt{10}i## is in the wrong quadrant. ##\text{arg}(w)=-\pi+\frac{\pi}{4}##.
  27. S

    Complex number problem: If arg(w)=π4 and |w⋅¯w|=20, then what is w of the form a+bi

    Great thanks. I get ##a=b= \pm\sqrt{10}## so ##w## follows.
  28. S

    Complex number problem: If arg(w)=π4 and |w⋅¯w|=20, then what is w of the form a+bi

    1. Homework Statement If ##\text{arg}(w)=\frac{\pi}{4}## and ##|w\cdot \bar{w}|=20##, then what is ##w## of the form ##a+bi##. 2. Homework Equations 3. The Attempt at a Solution The only way for the argument of ##w## to be ##\frac{\pi}{4}## is when ##a+bi## where ##a=b \in \mathbb{Z}##...
  29. S

    Double Integral question

    My solution is ##-\frac{106}{3}## which is clearly wrong as it has to be positive. My limits of integration on ##y## is ##0## and ##-2## for ##R_1## and ##0## and ##2## for ##R_2##.
Top