1. Inner product and angle in $\mathbb R^3$
Enter two vectors. The tool computes $x\cdot y$, norms, and the angle.
$x$
$y$
2. Projection onto a line in the plane
Orange is $y$. Blue is the line $L=\operatorname{span}\{w\}$. Black is $\operatorname{proj}_L(y)$. Red is the residual.
3. Gram--Schmidt and QR
Enter a $3\times2$ matrix $A=[b_1\ b_2]$. The tool computes an orthonormal basis and $A=QR$.
4. Least squares by QR
Fit $b\approx c_0+c_1t$ using four data points. This is projection onto the column space of $A=[1\ t]$.
5. Orthogonal matrix test
A matrix is orthogonal when $U^TU=I$. Orthogonal matrices preserve lengths and angles.
6. Complex adjoint
For complex matrices, the adjoint is $A^*=\overline A^T$, not just $A^T$.
Use entries $a+bi$.
7. Independent-study practice tasks with answers
Task A. For $x=(1,2,-1)$ and $y=(3,0,4)$, compute the angle.
Answer: $x\cdot y=-1$, $\|x\|=\sqrt6$, $\|y\|=5$, so $\cos\theta=-1/(5\sqrt6)$.
Answer: $x\cdot y=-1$, $\|x\|=\sqrt6$, $\|y\|=5$, so $\cos\theta=-1/(5\sqrt6)$.
Task B. Project $y=(4,1,2)$ onto $\operatorname{span}\{(1,2,0)\}$.
Answer: The projection is $(6/5,12/5,0)$ and the residual is $(14/5,-7/5,2)$.
Answer: The projection is $(6/5,12/5,0)$ and the residual is $(14/5,-7/5,2)$.
Task C. Apply Gram--Schmidt to $(1,1,1)$ and $(1,0,1)$.
Answer: An orthonormal basis is $(1,1,1)/\sqrt3$ and $(1,-2,1)/\sqrt6$.
Answer: An orthonormal basis is $(1,1,1)/\sqrt3$ and $(1,-2,1)/\sqrt6$.
Task D. Why is QR preferred for least squares?
Answer: QR avoids explicitly forming $A^TA$, which squares the condition number and can magnify numerical errors.
Answer: QR avoids explicitly forming $A^TA$, which squares the condition number and can magnify numerical errors.
Task E. Compute $A^*$ for $A=\begin{bmatrix}1&i\\2&1-i\end{bmatrix}$.
Answer: $A^*=\begin{bmatrix}1&2\\-i&1+i\end{bmatrix}$.
Answer: $A^*=\begin{bmatrix}1&2\\-i&1+i\end{bmatrix}$.