# Math-Magic Tree empty

by Yu · Published · Updated

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- 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 chosen 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 […]
- Determine a Condition on $a, b$ so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}\] be vectors in $\R^3$. Determine a […]
- Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by \[ T(X)=AX-XA\] for each $X\in V$. (a) Prove that $T:V\to V$ is a linear transformation. (b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]
- Prove the Cauchy-Schwarz Inequality Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$. Prove the Cauchy-Schwarz inequality: \[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\] We give two proofs. Proof 1 Let $x$ be a variable and consider the length of the vector […]
- The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$ Suppose that $p$ is a prime number greater than $3$. Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$. (b) Determine the index $[G : S]$. (c) Assume […]
- Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
- If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that $A$ is a real $n\times n$ matrix. (a) Is it true that $A$ must commute with its transpose? (b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set. Is it true that the rows of $A$ must also form an orthonormal set? (University of […]