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Không gian vector
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gacon Offline
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Không gian vector
Tổ hợp tuyến tính

1. Cho $u=(1,2,3), u_1=(1,0,1),u_2=(1,1,0),u_3=(0,1,1) \in\mathbb{R}^3$, vector $u$ có phải là tổ hợp tuyến tính của $u_1, u_2,u_3$ không?

2. Cho $u=(-2,-2,1), u_1=(2,-1,3),u_2=(4,1,2),u_3=(6,0,5) \in\mathbb{R}^3$, vector $u$ có phải là tổ hợp tuyến tính của $u_1, u_2,u_3$ không?

3. Cho $u=(2,-3,3), u_1=(1,-2,3),u_2=(0,1,-3) \in\mathbb{R}^3$, vector $u$ có phải là tổ hợp tuyến tính của $u_1, u_2$ không?

4. Trong $\mathbb{R}^3$ cho các vector $u_1=(1,-2,3), u_2=(0,1,-3)$. Xét xem vector $u=(2,-3,-3)$ có phải là tổ hợp tuyến tính của $u_1, u_2$ không?

5. Trong không gian các ma trận thực vuông cấp 2 $M_2(\mathbb{R})$, cho 4 vector $u=\begin{pmatrix}1&3\\ 2&2\end{pmatrix}, u_1=\begin{pmatrix}1&0\\ 1&0\end{pmatrix}, u_2=\begin{pmatrix}1&1\\ 0&0\end{pmatrix}, u_3=\begin{pmatrix}0&1\\ 1&1\end{pmatrix}$. Hỏi vector $u$ có là tổ hợp tuyến tính của $u_1,u_2,u_3$ không?

10. Trong $\mathbb{R}^3$, cho các vector $u=(1,m,-3), u_1=(1,-2,3), u_2=(0,1,-3)$. Tìm $m$ để $u$ là tổ hợp tuyến tính của $u_1,u_2$.

11. Tìm $m$ để vector $x=(7,-2,m)$ là tổ hợp tuyến tính của các vector $u=(2,3,5), v=(3,7,8), w=(1,-6,1)$.

12. Tìm $m$ để vector $x=(9,12,m)$ là tổ hợp tuyến tính của các vector $u=(3,4,2), v=(6,8,7)$.

13. Tìm $m$ để vector $x=(5,9,m)$ là tổ hợp tuyến tính của các vector $u=(4,4,3), v=(7,2,1), w=(4,1,6)$.

14. Tìm $m$ để vector $x=(1,3,5)$ là tổ hợp tuyến tính của các vector $u=(3,2,5), v=(2,4,7), w=(5,6,m)$.

Phụ thuộc tuyến tính - Độc lập tuyến tính

1. Trong $\mathbb{R}^3$, hệc vector $\{u_1=(1,1,0), u_2=(0,1,1), u_3=(1,0,1)\}$ có độc lập tuyến tính không?

2. Trong $\mathbb{R}^3$, hệc vector $\{ u_1 = (1,0,1);u_2 = (1,2,3);u_3 = (10,11,12);u_4 = (4,5,6)\}$ có độc lập tuyến tính không?

3. Trong $\mathbb{R}^3$, hệc vector $\{ u_1 = (1,0,1);u_2 = (-1,3,3);u_3 = (2,2,4)\}$ có độc lập tuyến tính không?

4. Trong $\mathbb{R}^3$, hệc vector $\{ u_1 = (1,-2,1);u_2 = (5,2,3);u_3 = (-2,2,-5)\}$ có độc lập tuyến tính không?

5. Trong $\mathbb{R}^4$, hệc vector $\{ u_1 = (1, - 2,3, - 4);u_2 = (3,3, - 5,1);u_3 = (3,0,3, - 10)\} $ có độc lập tuyến tính không?

6. Trong $\mathbb{R}[x]$(không gian các đa thức hệ số thực bậc không quá 3), các hệ vectơ sau có độc lập tuyến tính không?

$\hspace{2cm}a. \{ u_1 = x^3 - 2x + 3; u_2 = x^2 + 1\}$

$\hspace{2cm}b. \{ u_1 = x^3 - 2x + 3; u_2 = x^2 + 1; u_3 = 2x^3 + x^2 - 4x + 10\}$

$\hspace{2cm}c. \{ u_1 = x^3 - 2x + 3; u_2 = x^2 + 2x - 1; u_3 = x^3 + x^2 + 2\}$

$\hspace{2cm}d. \{ u_1 = x^3; u_2 = 2x^2; u_3 = 3x; u_4 = 2x^2 + 3x; u_5 = 1\}$

7. Chứng minh rằng hệ vector $\{u_1=(1,2,0,4), x_2=(-1,0,5,1), x_3=(1,6,10,14)\}$ phụ thuộc tuyến tính.

8. Cho $V$ là không gian vector trên $\mathbb{R}$ và $x, y, x \in V$. Chứng minh rằng $\{x, y, z\}$ độc lập tuyến tính khi và chỉ khi $\{ x+y, y+z, z+x\}$ cũng độc lập tuyến tính.

Tập sinh

1. Hệ vector $\{u_1=(1,1,1), u_2=(2,2,0), u_3=(3,1,0) \}$ có sinh ra $\mathbb{R}^3$ không?

2. Hệ vector $\{u_1=(2,-1,3), u_2=(4,1,2), u_3=(8,-1,8) \}$ có sinh ra $\mathbb{R}^3$ không?

3. Hệ vector $\{u_1=(2,-1,4), u_2=(3,-2,3), u_3=(4,3,5) \}$ có sinh ra $\mathbb{R}^3$ không?

4. Hệ vector $\{u_1=(3,1,4), u_2=(2,-3,5), u_3=(5,-2,9), u_4(1,4,-1) \}$ có sinh ra $\mathbb{R}^3$ không?

5. Hệ vector $\{u_1=(1,3,3), u_2=(1,3,4), u_3=(1,4,3), u_4(6,2,1) \}$ có sinh ra $\mathbb{R}^3$ không?

6. Hệ vector $\{u_1=(1,3,2), u_2=(2,-1,3), u_3=(3,-5,4), u_4(1,17,4) \}$ có sinh ra $\mathbb{R}^3$ không?

7. Hệ vector $\{u_1=(3,1,4), u_2=(2,-3,5), u_3=(5,-2,9), u_4(1,4,-1) \}$ có sinh ra $\mathbb{R}^3$ không?

8. Hệ vector $\{u_1=(1,2,3), u_2=(2,4,6), u_3=(3,6,9), u_4(2,8,12) \}$ có sinh ra $\mathbb{R}^3$ không?

9. Hệ vector $\{u_1=(1,3,3), u_2=(1,3,4), u_3=(1,4,3), u_4(6,2,1) \}$ có sinh ra $\mathbb{R}^3$ không?

10. Hệ vector $\{u_1=(1,2,2), u_2=(2,4,4), u_3=(-1,-2,-4), u_4(3,4,7) \}$ có sinh ra $\mathbb{R}^3$ không?


Cơ sở số chiều - Tọa độ của vector

2. Chứng minh $\mathfrak{B}=\{u_1 = (1,1,1);u_2 = (1,1,2);u_3 = (1,0,3)\}$ là cơ sở của $\mathbb{R}^3$, cho $u=(6,9,11)$ tìm $[u]_\mathfrak{B}$.

3. Chứng minh $\mathfrak{B}=\{u_1 = (2,1, - 3);u_2 = (3,2, - 5);u_3 = (1, - 1,1)\}$ là cơ sở của $\mathbb{R}^3$, cho $u = (6,2, - 7)$ tìm $[u]_\mathfrak{B}$.

4. Chứng minh $\mathfrak{B}=\{u_1 = (1, - 1,0);u_2 = (1,0, - 1);u_3 = (2,0,0)\}$ là cơ sở của $\mathbb{R}^3$, cho $u = ( - 3,1, - 2)$ tìm $[u]_\mathfrak{B}$.

5. Cho $\mathfrak{B}=\{u_1=(1,-2,3),u_2=(2,-1,5),u_3=(3,-4,16)\}$ Chứng minh $\mathfrak{B}$ là một cơ sở của $\mathbb{R}^3$. Cho $x=(2,4,6)$, Tìm $[x]_{\mathfrak{B}}$ - tọa độ của vector $x$ trong cơ sở $\mathfrak{B}$.

6. Cho $\mathfrak{B}=\{u_1=(1,2,-1,2),u_2=(2,3,0,-1),u_3=(1,2,1,3),u_4=(1,3,-1,0)\}$ Chứng minh $\mathfrak{B}$ là một cơ sở của $\mathbb{R}^4$.

7. Cho $\mathfrak{B}=\{u_1=(1,-2,3),u_2=(2,-1,5),u_3=(3,-4,16)\}$ Chứng minh $\mathfrak{B}$ là một cơ sở của $\mathbb{R}^3$.

8. Cho $\mathfrak{B}=\{u_1=(0,1,1,1),u_2=(1,0,1,1),u_3=(1,1,0,1),u_4=(1,1,1,0)\}$ Chứng minh $\mathfrak{B}$ là một cơ sở của $\mathbb{R}^4$. Tìm tọa độ của các vector $x=(1,1,1,1), y=(1,0,0,0)$ trong cơ sở $\mathbb{B}$.

9. Trong $\mathbb{R}^3$ gọi $\mathbb{B}$ là cơ sở chính tắc và $\mathbb{B}'$. Tìm ma trận $S$ là ma trận chuyển cơ sở từ $\mathbb{B}$ sang $\mathbb{B}'$ và ma trận chuyển cơ sở từ $\mathbb{B}'$ sang $\mathbb{B}$.


Ma trận chuyển cơ sở

1. Tìm ma trận chuyển cơ sở $P_{\mathbb{B}_1\to\mathbb{B}_2}$ từ cơ sở $\mathbb{B}_1= \{e_1=(-1,1,2), e_2=(1,2,1), e_3=(1,-2,3) \}$ sang cơ sở $\mathbb{B}_2=\{v_1=(-1,3,-2), v_2=(2,1,-1),v_3=(0,1,0)\}$.

2. Tìm ma trận chuyển cơ sở $P_{\mathbb{B}_1\to\mathbb{B}_2}$ từ cơ sở $\mathbb{B}_1= \{e_1=(1,2,4), e_2=(1,-1,1), e_3=(2,2,4) \}$ sang cơ sở $\mathbb{B}_2=\{v_1=(2,-7,1), v_2=(1,2,5),v_3=(1,3,2)\}$.

3. Tìm ma trận chuyển cơ sở $P_{\mathbb{B}_1\to\mathbb{B}_2}$ từ cơ sở $\mathbb{B}_1= \{e_1=(1,1,1), e_2=(1,1,2), e_3=(1,2,3) \}$ sang cơ sở $\mathbb{B}_2=\{v_1=(1,0,-1), v_2=(1,2,1),v_3=(0,-3,2)\}$.

4. Tìm ma trận chuyển cơ sở $P_{\mathbb{B}_1\to\mathbb{B}_2}$ từ cơ sở $\mathbb{B}_1= \{e_1=(1,1,2,1), e_2=(1,2,3,1), e_3=(1,3,5,2), e_4=(1,4,9,4) \}$ sang cơ sở $\mathbb{B}_2=\{v_1=(2,3,3,3), v_2=(-1,3,-1,-1),v_3=(3,3,-1,3), v_4=(2,2,-2,-2)\}$.
(This post was last modified: 10-31-2013 10:12 AM by gacon.)
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