# Math-Magic Tree empty

by Yu · Published · Updated

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

### More from my site

- The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic. The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chose the numbers $7, 5, 3, 2, […]
- True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample. Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$. If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a […]
- Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\] are linearly independent over $\R$. Hint. Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\] […]
- Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis. Determine the formula for the function $F$ and prove that $F$ is a linear transformation. Solution 1. Let $\begin{bmatrix} x \\ y […]
- Subset of Vectors Perpendicular to Two Vectors is a Subspace Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\] Prove that the subset $W$ is a subspace of […]
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
- Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors) Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation \[a_{k+2}-5a_{k+1}+3a_{k}=0\] for $k=1, 2, \dots$. Let $T$ be […]
- Sherman-Woodbery Formula for the Inverse Matrix Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies \[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\] Define the matrix […]