Solution

To find t for which the list (3, 1, 4), (2, −3, 5), (5, 9, t) is linearly independent, we can write the last vector (5, 9, t) as a linear combination of the first two vectors (3, 1, 4), (2, −3, 5) as follows:

(5, 9, t) = a₁(3, 1, 4) + a₂(2, −3, 5)

for some a₁, a₂ ∈ F and solve for t. To do this, we can rewrite the above equation as

(5, 9, t) = (3a₁ + 2a₂, a₁ − 3a₂, 4a₁ + 5a₂),

from which we can equate the coordinates of both sides to obtain the system of equations

5 = 3a₁ + 2a₂,

9 = a₁ − 3a₂,

t = 4a₁ + 5a₂

The first two of the three equations in the above system can be system-solved to get a₁ = 3,

Solution

Solving (1), (2) and (3) by using your calculator fx991Ms

we get a₁ = a₂ = a₃ = 0

Hence, the set is linearly independent

Solution

Solution

Solution

Solution

A is just the matrix product of L and U, so

Solution

The area of a triangle is given by

Solution

Here we have

From the last matrix in Step 2, we deduce row-rank(A) = 2

Solution

Applying addition and scalar multiplication in R5 to the left-hand side of the above equation, we get

from which we can equate the second, fourth, and fifth coordinates of both sides to obtain

and so the list (3,1,0,0,0), (0,0,7,1,0), (0,0,0,0,1) is linearly independent in U. So this list is a basis of U.

Solution

Hence, u1, u2, u3, u4 are linearly dependent.

Solution

⇒ α = 2 and β = 7.

Solution

Solution

Given λ₁ = 3 λ₂ = 15 λ₃ = ?

Solution

Solution

Solution

We apply Gaussian elimination. To keep track of the operations, we use, e.g., R₂ = R₂ − 2 ∗ R₁, which means that the new row 2 is computed by subtracting 2 times row 1 from row 2

The original system (1) is equivalent to

The system (2) can be solved by back-substitution. We get the solution

x₃ = 8/4 = 2,

x₂ = (4−1*2)/1 = 2,

x₁ = (2−4*2−(−2)*2)/2 = −1.

Solution

Normalization refers to organizing data to minimize redundancy and dependency. In databases, this is achieved by breaking down larger tables into smaller ones and linking them using relationships. This helps in reducing anomalies in database operations like insertions, deletions, and updates.

Solution

Systematic sampling selects every kth record from a population. The sampling interval k is determined by N/n​, where N is the population size (5000), and n is the sample size (500). Thus,

k=5000/500=10

Every 10th record is selected in the sample.

Solution

In a snowflake schema, dimension tables are normalized to reduce redundancy. This leads to multiple related tables for a single dimension, representing hierarchical levels, unlike the star schema where dimension tables are denormalized.

Solution

Derived measures are computed using one or more base measures. Examples of derived measures include "Sales growth percentage" and "Average sales per product." The number of products sold is a base measure, not derived from other measures.

Solution

The total number of rows in the fact table can be calculated as:

Rows of Products of Stores of DaysTotal Rows=Number of Products×Number of Stores×Number of Days

Substituting the values:

RowsTotal Rows=1,000×100×30=3,000,000

Therefore, the fact table will contain 3,000,000 rows after 30 days.

Solution

Min-max normalization is calculated using the formula:

Normalized value=(Xmax​−Xmin​)/(X−Xmin​)​

Here, X=60, Xmin​=1, and Xmax​=100:

Normalized value=(100−1)(60−1)​=9959​≈0.596≈0.6

Solution

The range of the data is 20−2=18. To divide it into 4 equal-width bins, the width of each bin is:

Solution

Solution

The number of cells in a fully populated data cube is the product of the unique values in each dimension. If each of the 4 dimensions has 10 unique values, the total number of cells is:

10×10×10×10 = 10,000

Solution

Solution

Combination of prime and non prime attributes should be taken as non prime only. Hence, in {wx->yz, xy->z} the FD xy->z is transitive dependency.

Solution

(a, a, c) - (a,c) = 0

Solution

The SQL LIKE clause is used to compare a value to similar values using wildcard operators. There are two wildcards used in conjunction with the LIKE operator.

• The percent sign (%)
• The underscore (_)

The percent sign represents

Solution

In D->C both C and D are non prime attributes. So, it is a transitive dependency but not partial dependency. Hence, D->C is in 2NF but not in 3NF

Solution

Solution

R2/R1 gives minimum of zero rows as there is no row in R2 for each row of R1.

R2/R1 gives max of X2/X1 rows as there is a row in R2 for each row of R1.

Solution

• Each entity in X is participating in one entity in relation.
• Each entity in Y is participating in one or more entities in relation.

We can merge X and Relation as one table: